THE ABSTRACT BOOK of the Conference ICAAM 2018

Fourth International Conference on Analysis and Applied Mathematics

THE ABSTRACT BOOK of the conference ICAAM 2018

Edited by Prof. Dr. Charyyar Ashyralyyev and Assoc. Prof. Dr. Evren Hincal

6-9 September, 2018

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey

1 Fourth International Conference on Analysis and Applied Mathematics 2

Fourth International Conference on Analysis and Applied Mathematics (ICAAM 2018): THE ABSTRACT BOOK / Edited by Charyyar Ashyralyyev and Evren Hincal, 2018. -177 pages.

ISBN 978-605-9415-37-8

The conference is organized biannually. Previous conferences were held in Gu- mushane, Turkey in 2012, in Shymkent, Kazakhstan in 2014, and in Almaty, Kazakhstan in 2016. The aim of the International Conference on Analysis and Applied Mathematics (ICAAM) is to bring mathematicians working in the area of analysis and applied mathematics together to share new trends of applications of mathematics. In mathematics, the developments in the ﬁeld of applied mathematics open new research areas in analysis and vice versa. That is why, we plan to found the conference series to provide a forum for researches and scientists to communicate their recent developments and to present their original results in various ﬁelds of analysis and applied mathematics. Web page: http://icaam-online.org/

ISBN 978-605-9415-37-8 c Fourth International Conference on Analysis and Applied Mathematics ICAAM 2018

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 3

Conference Sections and Minisymposiums : • Analysis • Applied Mathematics • Mathematics Education • Other Topics (Algebra, Geometry, Topology...) • Minisymposium:Functional analysis in interdisciplinary applications • Minisymposium: • Minisymposium:Actual mathematical problems in control theory • Poster Presentations Invited Speakers • Prof. Zuhair Nashe Modern Problems of Mathematical Hydrodynamics d, Uni- versity of Central Florida, USA • Prof. Mokhtar Kirane, University of La Rochelle, France • Prof. Michael Ruzhansky, Imperial College London, UK • Prof. Ljubisa D.R. Kocinac, University of Nis, Serbia • Prof. Alexey Lukashov, Moscow Institute of Physics and Technology, Russia • Prof. Osman Yurekli, Ithaca College, USA • Prof. Agacik Zafer, American University of the Middle East, Kuwait • Prof. Tynysbek Kalmenov, Institute of Mathematics and Mathematical Mod- elling, Kazakhstan • Prof. Akylbek Kerimbekov, Kyrgyz-Russian Slavic University, Kyrgyzstan • Prof. Stanislav Kharin, Kazakh-British Technical University, Kazakhstan

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 4 COMMITTEES HONORARY CHAIRS • Dr. Suat Í. Günsel,Founder Rector of Near East University, Turkey • Prof. Umit¨ Hassan, Rector of Near East University, Turkey • Prof. Ílkay Saliho˘glu,Rector of Girne University, Turkey • Emeritus Prof. Pavel E.Sobolevskii, Hebrew University, Israel CHAIRS • Prof. Allaberen Ashyralyev, Near East University, Turkey & Turkmenistan • Prof. Tynysbek Kalmenov, Institute of Mathematics and Mathematical Mod- elling, Kazakhstan CO-CHAIRS • Assoc. Prof. Abdullah S. Erdogan, Sigma LABS, Satbayev University, Kaza- khstan • Prof. Makhmud Sadybekov, Institute of Mathematics and Mathematical Mod- elling, Kazakhstan COORDINATING COMMITTEE • Prof. S¸enol Bekta¸s,Near East University, Turkey • Prof. Fahreddin M. Sadıko˘glu,Near East University, Turkey • Prof. Tamer S¸anlıda˘g,Near East University, Turkey • Assoc. Prof. Mustafa Kurt, Near East University, Turkey INTERNATIONAL ADVISORY BOARD • Prof. Ljubisa D.R. Kocinac, University of Nis, Serbia • Prof. Sergey Igorevich Kabanikhin, Novosibirsk State University, Russia • Prof. Mukhtarbay Otelbaev, Institute of Mathematics and Mathematical Mod- elling, Kazakhstan • Prof. Mokhtar Kirane, University of La Rochelle, France • Prof. Eberhard Malkowsky, State University Novi Pazar, Serbia • Prof. Valry C. Covachev, Institute of Mathematics and Informatics, Bulgaria • Prof. Osman Yurekli, Ithaca College, USA • Prof. Mohammad Sal Moslehian, Ferdowsi University of Mashhad, Iran • Prof. Mohammad Mursaleen, Aligarh Muslim University, India • Prof. Zuhair Nashed, University of Central Florida, USA • Prof. Sandra Pinelas, Academia Militar, Portugal • Prof. Michael Ruzhansky, Imperial College London, UK • Prof. Ravshan R. Ashurov, Institute of Mathematics of Academy of Science, Uzbekistan • Prof. Victor G. Zvyagin, Voronezh State University, Russia • Prof. Yury Shestopalov, Karlstad University, Sweden • Prof. Ihor Lubashevsky, University of Aizu, Japan • Prof. Stanislav Kharin, Kazakh-British Technical University, Kazakhstan • Prof. Valery Sergejevich Serov, University of Oulu, Finland • Prof. Sergey V. Astashkin, Samara State University, Russia • Prof. Alexey L. Lukashov, Moscow Institute of Physics and Technology, Russia • Prof. Virginia Kiryakova, Institute of Mathematics and Informatics, Bulgaria

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 5 INTERNATIONAL ORGANIZING COMMITTEE • Prof. Adg¨uzelDosiyev, Near East University, Turkey • Prof. Viktor Valentinovich Vlasov, Lomonosov Moscow State University, Russia • Assoc. Prof. Evren Hincal, Near East University,Turkey (Head of TPC) • Prof. Agamirza Bashirov, Eastern Mediterranean University, Turkey • Emeritus Professor Feyzi Basar, Inonu University, Turkey • Prof. Ludmila Stepanovna Pulkina, Samara State University, Russia • Prof. Abdizhahan Sarsenbi, M. Auezov South Kazakhstan State University, Kazakhstan • Prof. Charyyar Ashyralyyev, Gumushane University, Turkey • Prof. Misir Mardanov, IMM of National Academy of Sciences of Azerbaijan, Azerbaijan • Prof. Sergey Piskarev, Lomonosov Moscow State University, Russia • Prof. Vladimir Shaidurov, Siberian Federal University, Russia & Beihang Uni- versity, China • Prof. Hongyu Liu, Hong Kong Baptist University, Hong Kong • Prof. Oktay Mukhtarov, Gaziosmanpasa University, Turkey • Prof. Vladimir P. Orlov, Voronezh State University, Russia • Prof. Claudio Cuevas, Federal University of Pernambuco, Brazil • Prof. Mirsaid Aripov, National University of Uzbekistan, Uzbekistan • Prof. Martin Bohner, Missouri S&T University, USA • Prof. Pavel Pankov, International University of Kyrgyzstan, Kyrgyzstan • Prof. Reimund Rautmann, University of Paderborn, Germany • Prof. Mikhail G. Dmitriev, FRC ”Computer Science and Control” of RAS, Russia • Prof. Agacik Zafer, American University of the Middle East, Kuwait • Prof. Akylbek Kerimbekov, Kyrgyz-Russian Slavic University, Kyrgyzstan TECHNICAL PROGRAM COMMITTEE • Prof. Ali E. Ozkul,¨ Near East University,Turkey • Prof. Yagub A. Sharifov, Baku State University, Azerbaijan • Assoc. Prof. Deniz Agirseven, Trakya University, Turkey • Assoc. Prof. Maksat Ashyraliyev, Baheehir University, Turkey • Assoc. Prof. Zafer Cakir, Alanya Alaaddin Keykubat University, Turkey • Assoc. Prof. Okan Gercek, Girne American University, Turkey • Assoc. Prof. Durvudkhan Suragan, Nazarbayev University, Kazakhstan • Assoc. Prof. Behcet Oznacar,¨ Near East University,Turkey • Assoc. Prof. Andrei V. Zvyagin, Voronezh State University, Russia • Assoc. Prof. Ozgur Yildirim, Yildiz Technical University, Turkey • Asst. Prof. Yildirim Ozdemir, Duzce University, Turkey • Asst. Prof. Ali Denker, Near East University,Turkey • Asst. Prof. Medine Yeilkayagil, Usak University, Turkey • Postdoc Doghonay Arjmand, Ecole Polytechnique Federal de Lausanne, Switzer- land • Asst. Prof. Bilgen Kaymakamzade, Near East University,Turkey • Lecturer Dr. Isa Baba, Near East University,Turkey • Lecturer Dr. Kheireddine Belakroum, Frres Mentouri University Constantine, Algeria • Lecturer Farouk Saad Tijjani, Near East University,Turkey • Lecturer Dr. Ali Ugur Sazaklioglu, University of Turkish Aeronautical Associ- ation, Turkey

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 6

FOREWORD The Organizing Committee of ICAAM and Near East University, Lefkosa (Nicosia), Mersin 10, Turkey are pleased to invite you to the Fourth International Conference on Analysis and Applied Mathematics, ICAAM 2018. The meeting will be held on Septem- ber 6-9, 2018 in North Cyprus, Turkey. The conference is organized biannually. Previous conferences were held in Gu- mushane, Turkey in 2012, in Shymkent, Kazakhstan in 2014, and in Almaty, Kaza- khstan in 2016. The proceedings of ICAAM 2012, ICAAM 2014, and ICAAM 2016 were published in AIP (American Institute of Physics) Conference Proceedings. Near East University is pleased to host the fourth conference which is focused on various topics of analysis and its applications, applied mathematics and modeling. The conference will consist of plenary lectures, mini symposiums and contributed oral presentations. The proceedings of ICAAM 2018 published in AIP (American Institute of Physics) Conference Proceedings. Selected full papers of this conference will be published in peer-reviewed journals. The aim of the International Conference on Analysis and Applied Mathematics (ICAAM) is to bring mathematicians working in the area of analysis and applied mathematics together to share new trends of applications of mathematics. In mathematics, the developments in the ﬁeld of applied mathematics open new research areas in analysis and vice versa. That is why, we plan to found the conference series to provide a forum for researches and scientists to communicate their recent developments and to present their original results in various ﬁelds of analysis and applied mathematics. The Con- ference Organizing Committee would like to thank our sponsors. The main organizer of the conference is Near East University, Lefkosa (Nicosia), Mersin 10, Turkey. The conference is also supported by Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan. We would like to thank Near East University, Turkey and In- stitute of Mathematics and Mathematical Modeling, Kazakhstan for their support. We also would like to thank to all Invited Speakers, Co-Chairs, Coordinating Committee, In- ternational Organizing Committee, International Organizing Committee, and Technical Program Committee Members. With our best wishes and warm regards, Chairs: Prof. Allaberen Ashyralyev Prof. Tynysbek Kalmenov

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Contents

1 ANALYSIS 15 1.1 A note on some fixed point theorems for generalized expansive mappings in cone metric spaces over Banach algebras 16 1.2 Cone rectangular metric spaces over Banach algebras and fixed point results of T-contraction mappings 17 1.3 About of the spectrum of regular boundary value problems for one-dimensional differential operators 18 1.4 An Extended Generalized (p, q)-Extensions for the Apostol type Polynomials 19 1.5 On the Existence of Solutions for Fractional Differential Inclusions with Sum and Integral Boundary Conditions 19 1.6 On the Stablity of the solution of a certain nonlinear elliptic partial differential equations 20 1.7 Characteristic problems for a loaded hyperbolic equation with the wave operator in the principal part 21 1.8 Spectral Properties of one Elliptic Operator in a Punctured Domain 22 1.9 Multiperiodic Solutions of the Autonomous Systems with the Operator of Differentiation on the Lyapunov’s Vector Field 22 1.10 A generalized Banach contraction principle on cone pentagonal metric spaces over Banach algebras 23 1.11 Monotone operators and the proximal point algorithm in CAT (0) spaces 24 1.12 Generalized weighted composition operators between some spaces of analytic functions 24 1.13 On relativistic velocity addition 25 1.14 On solution uniqueness of the Cauchy problem for a third-order partial differential equation with time-fractional derivative 27 1.15 Existence and uniqueness results for Multiplicative Fractional differential equation with three point integral boundary value problem 28 1.16 Heisenberg’s inequality in Morrey spaces 29 2 Applied Mathematics 30 2.1 About convergence of difference approximations for optimization problems described by elliptic equations with mixed derivatives and unbounded non-linearity 31 2.2 Well-posedness and energy decay of solutions to a Lamésystem under boundary fractional derivative controls 32 2.3 On approximation of first order derivatives of complex-valued functions by finite differences 33 2.4 Some Parseval-Goldstein type identities for the generalized Macdonald and Hankel transforms 34 2.5 A study on Heron triangles and difference equations 35 2.6 An inverse problem of heat conduction in a degenerating domain 37 2.7 The Solution of One Dimensional Fractional Hyperbolic Partial Differential Equations Using Sumudu Transform 37 2.8 The Gauss Decomposition for the Group SL(2, R) and Casimir Operator 38 2.9 Investigation of the utility function in the problem of consumer choice 38

7 2.10 Travelling wave solutions for the two dimensional Hirota system of equations 39 2.11 Asymptotic expansion of the solution of the nonlocal boundary value problem with initial jumps for singularly perturbed integro-differential equation 40 2.12 Vertex magic total labeling of selected trees 41 2.13 Stability of basis property of a type of problems with nonlocal perturbation of boundary conditions 42 2.14 On the stability estimation of finite-difference analogue of the integral geometry problem with a weight function 43 2.15 Six Point Implicit Methods for the Pure Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation 44 2.16 Peculiarities of Constructing an Optimal Vector Distributed Control with Nonlinear Optimization of OscillationProcesses 44 2.17 Processing of big data in the detection of geochemical anomalies of rare-earth metal deposits 45 2.18 Iterative Method for Solving Non-Linear Navier-Stokes Equations in Complex Domains Taking Into Account Boundary Conditions with Uniform Accuracy 46 2.19 A third order of accuracy difference scheme for Bitsadze- Samarskii type multi-point overdetermined elliptic problem 47 2.20 Multi-Objective mean-CVaR model under VG process on evaluating Portfolio efficiency 48 2.21 Stationary boundary-value problems of the anisotropic elastic medium dynamics 49 2.22 On the system of rational difference equations 50 2.23 Special Functions and Heat Polynomials for the Solution of Free Boundary Problems 51 2.24 Numerical solution of stochastic mixed Volterra-Fredholm integral equations driven by space-time Brownian motion via two-dimensional block pulse functions 52 2.25 The construction of a solution of a related system of the Laguerre type 53 2.26 The Coinciding Polygonal Numbers and the Related Arithmetical Properties of Tchebyshef Polynomials 54 2.27 Construction of solutions of inhomogeneous systems of Jacobi type 55 2.28 Assessment of slope stability (on the example of Jalal-Abad in the area of Khamza street 56 2.29 A Discrete-time Holling Type II Model with Allee and Refuge Effects 57 2.30 On a first-order partial differential equation 58 2.31 Identification hyperbolic problems with nonlocal conditions 59 2.32 Numerical Solution of Nonlocal Elliptic Problems 60 2.33 Well-posedness of Neumann-type elliptic overdetermined problem with integral condition 61 2.34 On the numerical solution of advection-diffusion problems with singular source terms 62 2.35 A Second Order Difference Scheme for the Numerical Processing of Source Identification Terms in Hyperbolic Equations 63

8 2.36 Riesz basis property of generalized eigenfunctions for many interval BVTP’s with eigenparameter dependent boundary conditions 64 2.37 Lower bound estimation for eigenvalues for many interval BVP’s with eigenparameter dependent boundary conditions 65 2.38 Source Identification Problems for SchrödingerDifferential and Difference Equations 66 2.39 Stability of the third order partial differential equations with time delay 67 2.40 Stability of the third order partial differential equations with time delay 68 2.41 A Mathematical Cancer Model with BCG Immunotherapy Combined with Immune Checkpoints Inhibitors: an Optimal Control Approach 69 2.42 An Interval Based Social Accounting Matrix for a Regional Economy 70 2.43 An Input-Output Analysis for a Regional Economy: The Case of North Cyprus 71 2.44 Source Identification Problems for Two Dimensional Neutron Transport Differential and Difference Equations 72 2.45 New type Sturm-Liouville eigenvalue problem with interior singular point 73 2.46 New least-squares adjustments of nuclear binding energies of atomic nuclei 73 2.47 Adaptive conditional integrators sliding mode control scheme for a class of nonlinear systems 74 2.48 Review of modeling and simulation of natural nuclear reactors 75 2.49 Delay Epidemic Model with and without Vaccine 76 2.50 Principle of Independence of Continuation of Functions k- Significant Logic From Coding 77 2.51 Difference schemes for delay Schrödingerequation 78 2.52 On numerical solution of the Schrödinger-parabolic equations 78 2.53 On the numerical solution of nonlinear system of coupled sine-Gordon equations 79 2.54 A note on stable difference schemes for a third order partial differential equation 80 2.55 Invariant difference schemes for sine-Gordon equations 81 2.56 Unification of q−exponential function and related q−numbers and polynomials 82 2.57 Suitable variations for degenerate modes of the simplex method 83 2.58 Numerical solutions of source identification problem for hyperbolic-parabolic equations 85 2.59 On the stable difference scheme for the time delay telegraph equation 86 2.60 Numerical solutions of source identification problem for hyperbolic-parabolic equations 87 2.61 Numerical solutions of source identification problem for hyperbolic-parabolic equations 88 2.62 New phenomenas for degenerate nonlinear cross system with convective transfer and absorption 89 3 Mathematics Education 90

9 3.1 Problems of heat conduction with different boundary conditions in noncylindrical domains 91 3.2 Problems of heat conduction in noncylindrical domains with special boundary conditions containing the time derivative 92 3.3 The uniqueness of the solution of the two-dimensional direct problem of a wave process with an instantaneous source and a flat boundar 93 3.4 Numerical solution of a two-dimension direct problem of the wave process 94 3.5 If I Could Write Mathematics It Would be a Novel: A new Approach; Curiosity and What 95 3.6 The Misuse of Mathematical Proof Methods: New Approaches and Suggestions 96 3.7 A different critical approach to the teaching and obtaining of some elementary geometric inequalities 97 3.8 The Model of Career Application System in Distance Education 98 3.9 Mathematics through the Mathematical Modeling Method: The Geometrical Objects 99 4 Other Topics (Algebra, Geometry, Topology... 100 4.1 Existence and uniqueness of solutions for the first-order nonlinear integro-differential equations with three-point boundary conditions 101 4.2 Singularly perturbed first-order equations in complex domains that lose their uniqueness under degeneracy 102 4.3 Characterzaton of some types of compactness and constructon ndex compactness ≤ τ extensons by means of unform strutures 103 4.4 Strongly unformly paracompactness 104 4.5 Existence of immovability lines of a partial mapping of Euclidean space E59 105 4.6 Mean value of an arithmetic function associated with the Piltz divisor function 105 4.7 On the Menger algebras of quasi-open 106 5 Minisymposium: Functional analysis in interdisciplinary applications 107 5.1 Approximation of fractional semilinear Cauchy problem in a Banach spaces 108 5.2 Cordoba-Cordoba type inequality on homogenous Lie groups 109 5.3 Distribution of eigenvalues of a third-order differential operator with strongly regular boundary conditions 110 5.4 One class of inverse problems for reconstructing the process of heat conduction from nonlocal data 111 5.5 On coersive solvability of a third-order differential equation with oscillating intermediate coefficient 112 5.6 On a generalization of the Neumann problem for the Laplace equation 113 5.7 On Green’s function of certain boundary-value problems for the biharmonic equation 113 5.8 On the number of eigenvalues of correct restrictions and extensions for the Laplace operator 114 5.9 An expansion result for the equation of transverse vibration of two-layered composite 115 5.10 A tri-harmonic Neumann function for the unit disc 115

10 5.11 About the Generalized Dirichlet – Neumann Problem for an Elliptic Equation of High Order 116 5.12 On a certain class of pseudo-differential operators and related equations 117 5.13 Spectral Properties of the Sturm-Liouville Operator with a Parameter That Changes Sign and Their Usage to the Study of the Spectrum of Differential Operators of Mathematical Physics Belonging to Different Types 118 5.14 About Volterra’s nonlocal boundary value problem with displacement for the wave equation 119 5.15 On the Spectral Properties of a Wave Operator Perturbed by a Lower-Order Term 120 5.16 Compactness Of The Commutators Of Riesz Potential On Global Morrey-type Space 121 5.17 Sharp by order two-sided estimates for one class of function series 122 5.18 Degenerate operators for ordinary differential equations 123 5.19 Generalized three-dimensional singular integral equation by tube domain 124 5.20 On a boundary condition of Bitsadze-Samarskii for the Lavrent’ev-Bitsadze equation 125 5.21 Numerical Solution of a Two Dimensional Elliptic-Parabolic Equation with Dirichlet-Neumann Condition 126 5.22 On the unique solvability of a nonlocal problem with integral condition for the impulsive partial differential equation third order 127 5.23 On the spectral properties of a differential operator arising in the theory of Brownian motion 128 5.24 Duhamel principle for the time-fractional diffusion equation in unbounded domain 129 5.25 Criterion for the unconditional basicity of the root functions related to the second-order differential operator with involution 130 5.26 Compactness results for the resolvent of a singular second order hyperbolic operator with a mixed derivative 132 5.27 Structure of essential spectra and discrete spectrum of the energy operator of five-electron systems in the Hubbard model. Fifth doublet state 133 5.28 Reduction of Navier-Stokes Equation to a Linear Equation 134 5.29 Trigonometric system and optimal approximation 135 6 Minisymposium: Modern Problems of Mathematical Hydrodynamics 136 6.1 Weak solvability of some fractional viscoelasticity models 137 6.2 Investigation of the solvability of initial-boundary value problems for a viscoelastic model with memory 138 6.3 Trajectory and global attractors for termo-Voigt model 139 6.4 Average magnetic field in a random stream with an update 140 7 Minisymposium: Actual mathematical problems in control theory 141 7.1 Projector approach for constructing the zero order asymptotic solution for the singularly perturbed linear-quadratic control problem in a critical case 142 7.2 A Solution to Inverse Problems For Dynamic Control Systems 143

11 7.3 A design of an approximate regulator for a weakly nonlinear terminal control problem using the penalty functions method 144 8 Poster Presentations 145 8.1 A conjugate gradient method for global optimization via a random perturbations 146 8.2 Numerical simulation of magnetohydrodynamic process in metallurgical DC furnaces 147 8.3 Some approaches for numerical modeling of electrovortex flows in a finite volume 148 8.4 Nonstationary Growth of Spherical Particles in a Supercooled Melt 149 8.5 On the Theory of Two-Phase Zone Incipience due to the Effect of Constitutional Supercooling 150 8.6 Analytical Description of Directional Crystallization with Two-Phase Regions 151 8.7 Nonlinear Dynamics of Unsteady-State Crystallization of a Binary System 152 8.8 How the convective heat transport at the solid/liquid phase interface influences the stable mode of dendritic growth 153 8.9 Stability anlaysis of conformable fractional order nonlinear systems 153 8.10 A new sufficient conditions on the generalized spectrum method to deal with spectral pollution 154 8.11 On the Stability of High-Performance Crystal Growth Simulations by the MPFC Method 155 8.12 Algorithm of solution of a linear matrix inhomogeneous difference equation with small step 156 8.13 About one inverse problem of time fractional evolution with an involution perturbation 157 8.14 Canonical transformation and feedback linearization of state-dependent coefficient nonlinear systems 158 8.15 Strong approximation by Fourier series and generalized periodic Nikol’skij-Besov-Morrey spaces 159 8.16 Fourier method approach in mixed problems for the heat equation with involution perturbation 160 8.17 Modeling of quadratic programming of the mutual influence of elements of a dynamic series on examples of small agricultural formations 161 8.18 Development of the method and calculation of the heat transfer coefficient of the soil 162 8.19 On the Well-Posedness of the Nonlocal Boundary Value Problem for the Differential Equation of Elliptic Type 163 8.20 Error a posteriori estimation for a problem of optimal control of the obstacle 164 8.21 Time asymptotic behavior of exponential function of Sturm- Liouville operator on the line 165 8.22 On one class of problems for restoring the density of sources of the process of fractional heat conductivity with respect to initial and final temperature 166 8.23 Improving Performance of Indoor Visible Light Communication Systems with Hexagonal Designed LED Transmitters 167

12 8.24 Approximation of Maxwell’s microscopic boundary conditions for onedimensional nonstationary nonlinear Boltzmann’s twelve moment equations 168 8.25 The modeling of heat and mass transfer processes in colloidal suspension 169 8.26 On a parabolic equation with time delay 170 8.27 HAM Solution for a delay parabolic equation 171 8.28 The construction of a solution of a related system of the Laguerre type 172 8.29 Construction of solutions of inhomogeneous systems of Jacobi type 173 8.30 Numerical Modeling of a Liquid Droplet Surface Advection: Some Tests of the Modified VOF Algorithm 174 8.31 Two-dimensional calculations of stratified turbulent flow in a pipe 175 8.32 Solution of a linear problem of Florin type with the small parameters for the system of the parabolic equations 176 8.33 On construction of the set of comparison functions of the program motion in the probable statement 177 8.34 178

Fourth International Conference on Analysis and Applied Mathematics 15

ANALYSIS

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 16

A note on some ﬁxed point theorems for generalized expansive mappings in cone metric spaces over Banach algebras

Abba Auwalu Department of Mathematics Near East University, Nicosia–TRNC, Mersin 10, Turkey [email protected], [email protected] Abstract: In this paper, we prove some fixed point theorems for generalized expansive mappings in non-normal cone metric spaces over Banach algebras. Moreover, we give an example to elucidate our result. Our results are significant extension and generalizations of recent result of B. Jiang et al., (J. Comput. Anal. Appl., 21(6), 1103 - 1114, 2016) and many well-known results in the literature. The following is our main result which generalizes Theorem 3.1 in [1], Theorem 2.2 in [2], Theorem 2.1 in [3], and Theorems 2.3, 2.6 in [4]. Let (Y, ρ) be a complete cone metric space over Banach algebra A, a unit e and P be a solid cone in A. Let the mapping T : Y → Y be a surjective and satisfies the generalized expansive condition:

ρ(Ty, Tz) + ϑ1[ρ(y, Tz) + ρ(z, Ty)] < ϑ2ρ(y, z) + ϑ3ρ(y, Ty) + ϑ4ρ(z, Tz), −1 for all y, z ∈ Y , where ϑj ∈ P (j = 1, 2, 3, 4) such that (ϑ1 + ϑ2 − ϑ4) ∈ P, and a −1 spectral radius δ[(ϑ2 + ϑ3 − ϑ1) (e + ϑ1 − ϑ4)] < 1. Then T has a ﬁxed point in Y .

Keywords: Cone metric space over Banach algebras, generalized expansive mapping, ﬁxed point, c-sequence

2010 Mathematics Subject Classification: 47H10,54H25 References: [1] A. Ahmed and J.N. Salunke, Fixed point theorems of expanding mappings without continuity in cone metric space over Banach algebras, Int. J. Innov. Res. Sci. Engg. Tech., vol. 6, no. 1, 19–22, 2017. [2] B. Jiang, S. Xu, H. Huang and Z. Cai, Some fixed point theorems for generalized expansive mappings in cone metric spaces over Banach algebras, J. Comput. Anal. Appl., vol. 21, no. 6, 1103–1114, 2016. [3] X. Huang, C. Zhu and X. Wen, Fixed point theorems for expanding mappings in cone metric spaces, Math. Reports, vol. 14, no. 2, 141–148, 2012. [4] C.T. Aage and J.N. Salunke, Some fixed point theorems for expansion onto mappings on cone metric spaces, Acta Mathematica Sinica (English series), vol. 27, no. 6, 1101–1106, 2011.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 17

Cone rectangular metric spaces over Banach algebras and ﬁxed point results of T-contraction mappings Abba Auwalu1, Ali Denker2 1Department of Mathematics Near East University, Nicosia–TRNC, Mersin 10, Turkey [email protected], [email protected] 2Department of Mathematics Near East University, Nicosia–TRNC, Mersin 10, Turkey [email protected], [email protected]

Abstract: In this paper, we introduce the notion of T-contraction mappings on cone rectangular metric spaces over Banach algebras. Further, we establish the existence and uniqueness of ﬁxed point for such mappings. Our results extend and generalize the Ba- nach contraction principle and Edelstein ﬁxed point theorem given in [1–6], and many recent results in the literature. Moreover, an example to illustrate the main result is also presented.

Keywords: cone rectangular metric spaces, Banach algebras, T-contraction mapping, ﬁxed point.

2010 Mathematics Subject Classification: 47H10, 54H25 References: [1] L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, vol. 332, no. 2, 1468–1476, 2007. [2] A. Azam, M. Arshad, I. Beg, Banach contraction principle on cone rectangular metric spaces, Applicable Analysis and Discrete Mathematics, vol. 3, no. 2, 236–241, 2009. [3] A. Beiranvand, S. Moradi, M. Omid, H. Pazandeh, Two fixed point for special mappings, arxiv: 0903, 1504 v1 [math. FA], 2009. [4] H. Liu, S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Applications, vol. 2013, no. 320, 1–10, 2013. [5] A. Auwalu, Synchronal algorithm for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces, International Journal of Mathematical Analysis, vol. 8 no. 15, 727–745, 2014. [6] S. Shukla, S. Balasubramanian, M. Pavlovic, A generalized Banach fixed point theorem, Bulletin of the Malasiyan Mathematical Sciences Society, vol. 39, 1529–1539, 2016.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 18

About of the spectrum of regular boundary value problems for one-dimensional diﬀerential operators Tynysbek Kal’menov1, Nurbek Kakharman1,2 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, 2 Al-Farabi Kazakh National University, Kazakhstan [email protected], [email protected]

Abstract: In this paper, established that every regular non-volterra problem has an inﬁnite spectrum. On [0, b] we consider the general linear ordinary diﬀerential equation

(m) X (k) (1) LQu = u (x) + ak (x) u (x) = f (x) with generalized regular boundary conditions

(2) Q1u (0) + Q2u (b) = 0

where Q1 and Q2 are linear operators defined on the traces of the functions u (0) and u (b) and on the traces of its derivatives. It should be noted that the general form of the regular boundary conditions for equation (1) is given by Kalmenov and Otelbaev (2016). The question arises. Is there a regular boundary value problem for differential equations that has a finite spectrum? For the wide class of differential equations Kalmenov and Suragan (2008) established that if a regular boundary value problem has at least one eigenvalue, then the spectrum of this problem is infinite. m In this paper, we assume that a (x) ∈ C∞ [0, b], d a (x) ≤ M, x ∈ [0, b] , m = k dxm α 0, 1, 2, ..... established that every regular non-volterra problem (the spectrum consists of at least one point) has an infinite spectrum. This research is financially supported by a grant AP05133239 and by the target program BR05236656 from the Science Committee from the Ministry of Science and Education of the Republic of Kazakhstan.

Keywords: One-dimensional diﬀerential operators; regular boundary conditions; inﬁn- ity spectrum; eigenvalue

2010 Mathematics Subject Classiﬁcation: 34L10, 47E05, 34B09 References: [1] T.Sh. Kalmenov, M. Otelbaev. Boundary criterion for integral operators. Doklady Akademii Nauk, vol. 466, No. 4, 395398, 2016. [2] T. Sh. Kalmenov and D. K. Suragan, Dokl. Math.vol. 78, no. 3, 913-915, 2008.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 19

An Extended Generalized (p, q)-Extensions for the Apostol type Polynomials Ugur Duran1, Serkan Araci2, Mehmet Acikgoz3, Hemen Dutta4 1 Iskenderun Technical University, Turkey [email protected] 2 Hasan Kalyoncu University, Turkey [email protected] 3 Gaziantep University, Turkey [email protected] 4 Gauhati University, India hemen dutta08@rediﬀmail.com

Abstract: In this report, a new uniﬁed generalization for Apostol type polynomials of order α and level m based on (p,q)-numbers is introduced. Then, several new formulas and identities including some algebraic and diﬀerential properties are derived. Moreover, multifarious correlations including the well known (p,q)-polynomials such as Apostol type (p,q)-Bernoulli polynomials, Apostol type (p,q)-Euler polynomials, Apostol type (p,q)- Genocchi polynomials (p,q)-Stirling numbers of the second kind, and (p,q)-Bernstein polynomials. Along this note we mainly use techniques in the paper [1].

Keywords: (p,q)-numbers, Bernoulli polynomials, Euler polynomials, genocchi polynomials, Bernstein polynomials, Stirling numbers of the second kind, Cauchy product

2010 Mathematics Subject Classiﬁcation: 05A30, 11B68, 11B73. References: [1] L. Castilla, W. Ramirez, A. Urilies, An extended generalized q-extensions for the Apostol type polynomials, accepted for publication in Abstract and Applied Analysis, 2018.

On the Existence of Solutions for Fractional Diﬀerential Inclusions with Sum and Integral Boundary Conditions Djamila Seba Dynamic of Engines and Vibroacoustic Laboratory, University M’Hamed Bougara of Boumerd`es,Algeria djam [email protected]

Abstract: In this paper, by using the method of upper and lower solutions for multifunction, we investigate the existence of solutions for a boundary value problem for fractional diﬀerential inclusions with sum and integral boundary conditions.

Keywords: Caputo fractional derivative, Fractional diﬀerential inclusion, upper and lower solutions ,Boundary value problem, ﬁxed point, Banach space.

2010 Mathematics Subject Classification: 26A33, 34B15, 34G20 References: [1] Djamila Seba, Nonlinear Fractional Differential inclusion with Nonlocal Fractional Integro- Differential Boundary Conditions in Banach Spaces, Mathematica Bohemica (2017). .

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 20

On the Stablity of the solution of a certain nonlinear elliptic partial diﬀerential equations Zayid Abdulhadi

1 Department of Mathematics and Statistics American University of Sharjah, Sharjah, UAE [email protected] Abstract: This report is devoted to the functions which are solution of the non- f f linear elliptic partial diﬀerential equation z = a z deﬁned on the unit disc U, a f f analytic, and a(U) ⊂ U. Such functions are called logharmonic mappings.We study logharmonic mappings which are stable univalent, stable starlike, stable with positive real part , stable close to starlike and stable typically real. We prove that the map- λ pings fλ = zh(z)g(z) are starlike logharmonic(resp. logharmonic univalent , close to starlike logharmonic, typically real logharmonic) for all |λ| = 1 if and only if the map- zh pings ϕ = are starlike analytic (resp. analytic univalent, close to starlike analytic, λ gλ typically real analytic) for all |λ| = 1.

Keywords: Logharmonic mappings, stable, Starlike , Typicallyreal, univalent 2010 MSC: Primary 30C45, 30C55; Secondary 35Q30. References: [1] Z. Abdulhadi and D. Bshouty, Univalent functions in H · H(D), Trans. Amer. Math. Soc. 305 (1988), no. 2, 841–849. [2] Z. Abdulhadi and W. Hengartner, Polynomials in HH, Complex Variables Theory Appl. 46 (2001), no. 2, 89–107. [3] X. Chen and T. Qian, Non-stretch mappings for a sharp estimate of the Beurling-Ahlfors operator. J. Math. Anal. Appl. 412 (2014), no. 2, 805-815. [4] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3–25. [5] R. Hernandez and M. J. Martın, Stable geometric properties of analytic and harmoniC functions, Proc. Cambridge Phil. Soc., 155(2013), 343–359 [6] Zh. Mao, S. Ponnusamy, and X. Wang, Schwarzian derivative and Landau’s theorem for logharmonic mappings. Complex Var. Elliptic Equ. 58 (2013), no. 8, 1093-1107.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 21

Characteristic problems for a loaded hyperbolic equation with the wave operator in the principal part Anatoly Attaev Institute of Applied Mathematics and Automation, Russia [email protected]

Abstract: In this paper we consider an equation of the form

(1) uxx − uyy + a(x, y)ux + b(x, y)uy + c(x, y)u

n X αi ξ + α(ξ) α(ξ) − ξ = λ ai(x, y)D u , , 0ξ 2 2 i=1 l where ξ = x − y, λ is the real number, α1 < α2 < ... < αn = α < 1,Dax is the operator of fractional integration as l < 0 and fractional differentiation as l > 0 of order |l| with the starting point a and the end point x, α(ξ) is a C2-diffeomorphism, and moreover (2) 0 ≤ α(ξ) ≤ 1, ξ ∈ [0, 1], α(0) = 0, ϕ, ψ ∈ C(J¯), ϕ(0) = ψ(0), 1 (3) a, b, c ∈ C (Ω), ai ∈ C(Ω)¯ , i = 1, n, where J¯ is the closure of the interval J = {(x, y), 0 < x < 1, y = 0}. Assume Ω = {(x, y) : 0 < x + y < 1, 0 < x − y < 1} is the rectangular domain bounded by characteristic equation (1). Problem G. In the domain Ω find the solution to equation (1) belonging to C(Ω)¯ ∩ C2(Ω), satisfying the boundary conditions x x (4) u , = ψ(x), 0 ≤ x ≤ 1, 2 2 x x (5) u , − = ϕ(x), 0 ≤ x ≤ 1, 2 2 where Ω¯ is the closure of the domain Ω. Theorem. Under the assumptions (2) and (3), the Goursat problem (1), (4) and (5) is always uniquely solvable.

Keywords: hyperbolic equation, fractional derivative, Goursat problem, wave equation, loaded equation

2010 Mathematics Subject Classiﬁcation: 35L05

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 22

Spectral Properties of one Elliptic Operator in a Punctured Domain Gulzat Nalzhupbayeva1 1Institute of Mathematics and Mathematical Modeling, Kazakhstan, and Al-Farabi Kazakh National University, Kazakhstan [email protected] Abstract: In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the ﬁxed iterating order m ∈ N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ≤ d ≤ 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ’nonharmonic’ analysis.

Keywords: Regularized trace formula, m-Laplace operator, punctured domain, non– harmonic analysis

2010 Mathematics Subject Classiﬁcation: 35J05

Multiperiodic Solutions of the Autonomous Systems with the Operator of Differentiation on the Lyapunov’s Vector Field Z.A. Sartabanov1, B.Zh. Omarova1 1 K.Zhubanov Aktobe Regional State University, 34 A.Moldagulova avenue, Aktobe, 030000, Kazakhstan [email protected], [email protected] Abstract: In this article we consider an autonomous system of a special form with a operator of differentiation whose characteristic system is related with the Lyapunov’s systems. The multiperiodicity of zeros of this operator has established. By the methods of the theory of multiperiodic solutions, we prove a theorem on the existence of a unique multiperiodic solution with respect to time variables. The result of the theorem extends to the nonlinear case on the basis of the fixed point method. The content of the article was the development of a method for investigating the oscillatory solutions of autonomous systems of the form (1). The main essence of the method for investigating the multiperiodic solutions of the system under consideration is a combination of the known methods of [1,2], developed in [3]. In this case, the characteristic system of the operator of differentiation consists of several subsystems of the Lyapunov’s class, which generates the multyperiodicity of the solutions of the original system. Note that the developed technique is applicable to a more general system when it contains both time and space variables. The idea of a work is new, which in the simplest case was first tested in [4].

References: [1] A.M. Lyapunov, The general problem of the stability of motion GITTL, Moscow-Leningrad, 1950, (In Russian). [2] V. Kh. Kharasakhal, Almost-Periodic Solutions of Ordinary Differential Equations, Nauka, Alma- Ata, 1970, (in Russian). [3] A. A. Kulzhumieva, Zh. A. Sartabanov, Periodic solutions of the systems of differential equations with multidimensional time, RIC WKSU, Uralsk, 2013, (in Russian). [4] Zh. A. Sartabanov, B. Zh. Omarova, ”The Lyapunov method in the investigation of the multiperiodic solutions of the one linear system of equations with the operator of differentiation”, Taimanov Readings 2017, Proceedings of the International Scientific and Practical Conference, Uralsk, Kazakhstan, 2017, pp.67-72.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 23

A generalized Banach contraction principle on cone pentagonal metric spaces over Banach algebras Abba Auwalu1, Evren Hın¸cal2 1Department of Mathematics Near East University, Nicosia–TRNC, Mersin 10, Turkey [email protected], [email protected] 2Department of Mathematics Near East University, Nicosia–TRNC, Mersin 10, Turkey [email protected], [email protected]

Abstract: In this paper, we introduce the concept of cone pentagonal metric spaces over Banach algebras as a generalization of metric space and many of its generalization such as; cone metric space [1,2], cone rectangular metric space [3], and cone pentagonal metric space [4]. Furthermore, we prove a generalized Banach contraction principle in such a space as follows:

Theorem 1.1. Let (X , d) be a complete cone pentagonal metric space over Banach algebra B and S be a non normal solid cone in B. Suppose T : X → X is a mapping that satisﬁes the following condition: d(T x, T y) 4 kd(x, y) for all x, y ∈ X , where k ∈ S is a generalized Lipschitz constant such that the spectral radius δ(k) < 1. Then T has a unique ﬁxed point x∗ in X . Moreover, for any x ∈ X , the iterative i ∗ sequence {T x} (i ∈ N) converges to x .

Keywords: cone pentagonal metric spaces, Banach algebras, c-sequence, contraction mapping principle, ﬁxed point.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 24

Monotone operators and the proximal point algorithm in CAT (0) spaces

G. Zamani Eskandani1, M. Raeisi2 1 Faculty of Sciences, Department of Mathematics, University of Tabriz, Tabriz, Iran [email protected] 2 Faculty of Sciences, Department of Mathematics, University of Tabriz, Tabriz, Iran [email protected]

Abstract: In this paper, by using products of finitely many resolvents of monotone operators, we propose an iterative algorithm for finding a common zero of a finite family of monotone operators and a common fixed point of a finite family of nonexpansive mappings in CAT (0) spaces. We derive the strong convergence of the proposed algorithm under appropriate conditions.

Keywords: Proximal point algorithm, CAT (0) spaces, monotone operator

2010 Mathematics Subject Classiﬁcation: 47H09, 47H10 References: [1] B. A. Kakavandi, Weak topologies in complete CAT (0) metric spaces. Proc Am Math Soc. 141 ( 2013), 1029- 1039. [2] M. Baˇca´k, Convex Analysis and Optimization in Hadamard Spaces, volume 22 of De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter, Berlin, (2014). [3] S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 13 (2000), 226-240. [4] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877-898.

Generalized weighted composition operators between some spaces of analytic functions

Hamid Vaezi1, Ebrahim Abbasi2 1 Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran [email protected] 2 Faculty of Mathematical Sciences University of Tabriz, Tabriz, Iran [email protected]

Abstract: In this work, we give some estimates for the essential norm of generalized n weighted composition operator Dϕ,u from Bloch type spaces into weighted type spaces. Also we obtain new condition for boundedness and compactness of this operator.

Keywords: generalized weighted composition operators, Bloch type space, essential norm, weighted type spaces.

2010 Mathematics Subject Classiﬁcation: 30H30, 30H99

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 25

On relativistic velocity addition

Agamirza E. Bashirov1, Sinem Genc2

1 Department of Mathematics, Eastern Mediterranean University Gazimagusa, Mersin 10, Turkey and Institute of Control Systems, ANAS, Baku, Azerbaijan

[email protected] 2 Department of Mathematics, Eastern Mediterranean University Gazimagusa, Mersin 10, Turkey

[email protected]

Abstract: In Einstein’s relativity theory, the sum of the velocities v1 = hv1, 0, 0i and v2 = hv2, 0, 0i equals to v = hv, 0, 0i with v + v (1) v = 1 2 , 2 1 + v1v2/c where c is the speed of light. This is a special case when the velocities v1 and v2 have the same direction. In more general case, when v1 and v2 have diﬀerent directions, the formula is more complicated and does not have ordinary properties such as linearity, commutativity, associativity, etc. In this work we are aimed to develop algebraic operations over the preceding relativistic addition formula which have all ordinary properties.

The method that we will use is based on non-Newtonian calculi initiated in [1]. Previously, non-Newtonian calculus with the exponential reference function, which has the multiplicative and bigeometric modiﬁcations, was studied in [2, 3]. In this paper, a suitable non-Newtonian calculus is with the hyperbolic tangent reference function.

We will operate with relative speeds, that is, instead of v we will consider v/c as- suming that |v| < c. The relative speeds are numbers in E = (−1, 1). Then for relative speeds v1 and v2, Eq. (1) produces the relative speed v by

v1 + v2 v = v1 ⊕ v2 = . 1 + v1v2 This is an addition formula in E.

For other algebraic operations in E, consider hyperbolic tangent function and its inverse deﬁned by ex − e−x α(x) = tanh x = , x ∈ R, ex + e−x and −1 −1 1 1 + x α (x) = tanh x = ln , x ∈ E. 2 1 − x It is known that (see, for example, [4]) a + b (2) (i) a ⊕ b = α(α−1(a) + α−1(b)) = . 1 + ab

Based on this, introduce the other algebraic operations on E by

−1 −1 a−b (ii) a b = α(α (a) − α (b)) = 1−ab ,

1 ln 1+b 1+a 2 1−b −1 (iii) a ⊗ b = α(α−1(a) × α−1(b)) = 1−a , 1 ln 1+b 1+a 2 1−b 1−a +1 2 ln 1+b 1+a 1−b −1 −1 −1 1−a (iv) a b = α(α (a)/α (b)) = 2 . ln 1+b 1+a 1−b 1−a +1 The formulae for a ⊗ b and a b are so that it is difficult to predict them by intuition unless they are not transformed in accordance to hyperbolic tangent reference function. Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 26 The set E with the operations (i)-(iv) transforms all properties of R to E making it an ordered field. In particular, the distributivity property can be verified as follows: (a ⊕ b) ⊗ c = α(α−1(a) + α−1(b)) ⊗ c = α(α−1(α(α−1(a) + α−1(b))) × α−1(c)) = α((α−1(a) + α−1(b)) × α−1(c)) = α((α−1(a) × α−1(c) + α−1(b) × α−1(c)) = α((α−1(a) × α−1(c)) ⊕ α(α−1(b) × α−1(c)) = (a ⊗ c) ⊕ (b ⊗ c). Moreover, the vector operations (addition, product by scalar, scalar product, vector product, etc.), defined through operations (i)-(iv) of E have all properties of the same operations defined over R. Whole ordinary calculus transforms to a new calculus that we can call tanh-calculus, in which the sense of derivative and integral changes. If f ? denotes the tanh-derivative of f, then

2f02) α−1(f(x))0 e 1−f2(x) − 1 (3) f ?(x) = α = . α−1(x)0 2f02) e 1−f2(x) + 1 This can be Informally derived as follows: α−1(α(α−1(f(y)) − α−1(f(x)))) ((f(y) f(x)) (y x)) = α α−1(α(α−1(y) − α−1(x))) α−1(f(y)) − α−1(f(x)) = α α−1(y) − α−1(x) α−1(f(y)) − α−1(f(x)) y − x = α · y − x α−1(y) − α−1(x) α−1(f(x))0 → α . α−1(x)0 In a similar way, informally we have n n M X −1 −1 −1 f(ci) ⊗ (xi xi−1) = α α (f(ci))(α (xi) − α (xi−1)) i=1 i=1 n −1 −1 X −1 α (xi) − α (xi−1) = α α (f(ci))(xi − xi−1) x − x i=1 i i−1 Z b → α α−1(f(x))α−1(x)0 dx . a Therefore, the tanh-integral is deﬁned by

2 R b ln 1+f(x) dx Z b Z b e a (1−f(x))(1−x2) − 1 (4) f(x) d?x = α α−1(f(x))α−1(x)0 dx = . R b 1+f(x) a a 2 ln dx e a (1−f(x))(1−x2) + 1 A removal of α−1(x)0 in (3) and (4), produces another simplified modification of tanh- derivative and tanh-integral. References: [1] Grossman M, Katz R, Non-Newtonian Calculi. Lee Press: Pigeon Cove; 1972. [2] Bashirov AE, Kurpınar E, Ozyapıcı¨ A. Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications. 2008; 337 (1):36-48. DOI: 10.1016/j.jmaa.2007.03.081. [3] Bashirov AE, Norozpour S. On an alternative view to complex calculus, Mathematical Methods in the Applied Sciences. 2018; DOI: 10.1002/mma.4827, 1-12. [4] Bashirov AE, Mısırlı E, Tando˘gduY, Ozyapıcı¨ A. On modeling with multiplicative differential equations, Applied Mathematics, Journal of Chinese Universities, Ser. 2011; 26 (4):425-438.

Keywords: Relativistic velocity, calculus, derivative, integral

2010 Mathematics Subject Classiﬁcation: 30E20,30E99

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 27

On solution uniqueness of the Cauchy problem for a third-order partial diﬀerential equation with time-fractional derivative Arsen Pskhu Institute of Applied Mathematics and Automation, Russia [email protected]

Abstract: Consider the equation ∂σ ∂3 (1) − u(x, y) = f(x, y), ∂yσ ∂x3 where σ ∈ (0, 1), and ∂σ/∂yσ stands for the fractional derivative of order σ with respect to y. The fractional diﬀerentiation is given by the Dzhrbashyan-Nersesyan operator (see [1]) associated with ordered pair {α, β}, i.e. ∂σ (2) = D{α,β} = Dβ−1Dα , α, β ∈ (0, 1], σ = α + β − 1, ∂yσ 0y 0y 0y β−1 α where D0y and D0y are the Riemann-Liouville fractional integral and derivative, respectively (see [2]). In paper [3], a fundamental solution of equation (1) and a representation for solution of the Cauchy problem α−1 (3) lim D u(x, y) = τ(x), x ∈ R, y→0 0y in the domain R × (0,T ), were constructed. Here, we prove a uniqueness theorem for the problem (1), (3) in the class of fast- growing functions satisfying an analogue of the Tychonoﬀ condition.

Keywords: fractional derivative, third-order partial diﬀerential equation, Tychonoﬀ’s condition, Dzhrbashyan-Nersesyan operator, Cauchy problem

2010 Mathematics Subject Classiﬁcation: 35R11, 35A02 References: [1] M.M. Dzhrbashyan, A.B. Nersesyan, Fractional derivatives and the Cauchy problem for diﬀerential equations of fractional order, Izv. Akad. Nauk Armenian SSR Matem., vol. 3, no 1, 3–28, 1968. [2] A.M. Nakhushev, Fractional calculus and its applications, Fizmatlit, Moscow, 271, 2003. [3] A.V. Pskhu, Fundamental solution for third-order equation with fractional derivative, Uzbek Math- ematical Journal, no 4, 119–127, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 28

Existence and uniqueness results for Multiplicative Fractional diﬀerential equation with three point integral boundary value problem Sajedeh Norozpour1 1 Department of Mathematics, Near East University

Abstract: Fractional differential equations have been a hot topic of recent years since it found many applications ranging from Biology, Chemistry, Engineering, physics, control theory, Electrical circuits [2]. On the other hand, the concept of multiplicative calculus which was well established in [1] is another active research area since it provides an interesting aspects of the known phenomena of mathematics and provide important achievement in some areas such as Numerical Analysis, Fractional and complex analysis. Furthermore, this topic starts to find many applications in economics , Image processing and functional Analysis. Recently, the multiplicative fractional differentiation and integration have been defining in [3] and in this paper, we focus on the investigation of the existence and uniqueness of multiplicative fractional differential equation with three point boundary value problem below; ∗ q c D x(t) = f(t, x(t)) x(0) = 1 Z η x(1) = ( x(s)ds)α 0 the most important note is that Multiplicative fractional calculus is established for some differential equations with non-linear continuous function. Additionally the multiplicative analogue of the solution of BVP is defined.

Keywords: multiplicative, fractional calculus, three point BVP, Caputo derivative, diﬀerential equation. References: [1] Bashirov, A. E., Kurpinar, E. M., & Ozyapici, A. (2008). Multiplicative calculus and its applications. Journal of Mathematical Analysis and Applications, 337(1), 36-48. [2] Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional integrals and derivatives: theory and applications. [3] Jiang, W. (2011). The existence of solutions to boundary value problems of fractional diﬀerential equations at resonance. Nonlinear Analysis: Theory, Methods & Applications, 74(5), 1987-1994.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 29

Heisenberg’s inequality in Morrey spaces Denny Ivanal Hakim 1 1 Department of Mathematics, Bandung Institute of Technology, Indonesia [email protected]

Abstract: The (classical) Heisenberg inequality is given by Z 1 Z 1 2 2 2 2 2 ˆ 2 2 (1) kfkL2 . (x − a) |f(x)| dx (ξ − α) |f(ξ)| dξ , R R 2 where f ∈ L (R), a, α ∈ R, and fˆ denotes the Fourier transform of f. There are several n generalizations of (1) in Lebesgue spaces over R (see [1–4]). In this talk we discuss a generalization of the Heisenberg inequality in Morrey spaces. We combine some Hardy type inequality and an interpolation inequality for the fractional power of the Laplacian to prove our result and to remove some restrictions in [2]. Our proof follows the idea in [3]. This is a joint work with Hendra Gunawan (Bandung Institute of Technology), Eiichi Nakai (Ibaraki University), and Yoshihiro Sawano (Tokyo Metropolitan University) (see [5]).

Keywords: Imaginary power of Laplace operators, fractional power of Laplace operators, interpolation inequality, Hardy’s inequality, Heisenberg’s inequality, Morrey spaces.

2010 Mathematics Subject Classiﬁcation: 42B20,42B35 References: [1] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123, 1995. [2] W. Beckner, Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136, 2008. [3] P. Ciatti, M. Cowling, and F. Ricci, Hardy and uncertainty inequalities on stratiﬁed Lie groups, Adv. Math. 277, 2015. [4] G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3, 1997, 207–238. [5] H. Gunawan, D.I. Hakim, E. Nakai, and Y. Sawano, Hardy-type and Heisenberg’s inequality in Morrey spaces, Bull. Austral. Math. Soc. 2018.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 30

Applied Mathematics

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 31

About convergence of diﬀerence approximations for optimization problems described by elliptic equations with mixed derivatives and unbounded non-linearity Aigul Manapova1, Fedor Lubyshev2 1 Bashkir State University, Ufa, Republic of Bashkortostan, Russia [email protected] 2 Bashkir State University, Ufa, Republic of Bashkortostan, Russia

Abstract: This report is devoted to an optimal control problem for nonlinear elliptic equation with mixed derivatives and unbounded non-linearity, namely, minimizing of a Z 2 functional J(g) = |u(x; g) − u0(x)| dΩ, on solutions u(g) to problem Ω 2 X ∂2u ∂2u − kαα(x) − 2k12(x) + q(u)u = f(u), x ∈ Ω; ∂x2 ∂x ∂x α=1 α 1 2 u(x) = 0, x ∈ ∂Ω = Γ, obeying all admissible controls 1 g = k11, k22, k12 ∈ U = kαβ ≡ gαβ ∈ W∞(Ω), α, β = 1, 2 : 2 2 2 X 2 X X 2 ν ξα ≤ kαβ (x)ξαξβ ≤ µ ξα, kαβ (x) = kβα(x), α=1 α,β=1 α=1 2 ∂kαβ ∂kαβ ∀x ∈ Ω, ξ 6= 0, ξ ∈ R , ≤ R1, ≤ R2, α, β = 1, 2 , ∂x1 ∂x2 (1) 1 where u0 ∈ W2 (Ω1) is a given function, q(η), f(η) are given functions of η. Suppose that the conditions imposed on the coefficients of the state equation are satisfied only in the vicinity of the exact solution values that indicates the presence of non-linearities of the unbounded growth. Present work is dedicated to the development and study of difference approximations for the given class of optimization problems, namely, study of issues, such as the convergence of approximations with respect to the state, functional, and control (see, e.g., [1], [2] and the references therein).

Keywords: Elliptic equations with mixed derivatives, optimal control, objective functional, unbounded non-linearity, diﬀerence method

2010 Mathematics Subject Classification: 49J20, 35J61, 65N06 References: [1] F.V. Lubyshev, A.R. Manapova, “Difference approximations of optimization problems for semilinear elliptic equations in a convex domain with controls in the coefficients multiplying the highest derivatives”, Comput. Math. Math. Phys., Vol. 53, No.1, pp. 8–33, 2013. [2] Manapova, A.R., Lubyshev, F.V., “Accuracy estimate with respect to state of finite-dimensional approximations for optimization problems for semi-linear elliptic equations with discontinuous coefficients and solutions”, Ufa Mathematical Journal, Vol.6, No.3, pp. 69-84, 2014.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 32

Well-posedness and energy decay of solutions to a Lam´esystem under boundary fractional derivative controls Gaouar Soumia1, Benaissa Abbes2 1 Laboratory of Mathematics, Djillali Liabes University, Algeria [email protected] 2 Laboratory of Mathematics, Djillali Liabes University, Algeria

benaissa−[email protected]

Abstract: In this paper, we investigate the existence and decay properties of solutions for the initial boundary value problem of the Lam´esystem with boundary control conditions of fractional derivative type:

 u − µ∆u − (µ + λ)∇(div u) = 0 in Ω × (0, +∞)  tt (P ) u = 0 in Γ0 × (0, +∞)  ∂u α,η µ ∂ν + (µ + λ)(div u)ν = −γ∂t u in Γ1 × (0, +∞)

T where u = (u1, u2, ..., un) , ν stands for the unit normal vector of Ω pointing to- α,η wards the exterior and µ, λ, γ are positive constants. The notation ∂t stands for the generalized Caputo’s fractional derivative of order α with respect to the time variable. It is deﬁned as follows Z t α,η 1 −α −η(t−s) dw ∂t w(t) = (t − s) e (s) ds η ≥ 0. Γ(1 − α) 0 ds The system is ﬁnally completed with initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x).

Our purpose in this paper is to give a global solvability in Sobolev spaces and energy decay estimates of the solutions to the problem (P ).

Keywords: Lam´esystem, Fractional feedback, Polynomial stability, Semigroup theory.

2010 Mathematics Subject Classiﬁcation: 35B40, 74D05, 47D03 References: [1] A. Benaissa and S. Benazzouz. Well-posedness and asymptotic behavior of Timoshenko beam system with dynamic boundary dissipative feedback of fractional derivative type.Springer International Publishing AG, 2016; 10.1007. [2] Mbodje B. Wave energy decay under fractionel derivative controls. IMA Journal of Mathematical Control And information 2006; 23.237-257.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 33

On approximation of first order derivatives of complex-valued functions by finite differences Charyyar Ashyralyyev 1,1, Beyza Oztürk¨ 1,2 1,2Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey

E-mail: [email protected], [email protected]

Abstract: Boundary value problems for partial differential equations involving complex valued functions have important applications in a broad sense ( [1–5]). The theory of finite difference method in case of real valued function and its applications to solve boundary value problems for partial differential equations is described in [6]. Complex step method for computing derivatives of real valued functions by introducing a complex step in a strict sense is considered in [7, 8] (see also references therein). In this presentation, we generalize the well known finite difference method to compute derivatives of real valued function to approximate of complex derivatives wz and w z for complex valued function w. Exploring different combinations of terms, we derive several approximations to compute the first order derivatives of complex valued function w. The first, second, third and fourth order of accuracy finite differences to calculate derivatives are studied. Error analyses in test examples are carried out by using Matlab program.

Keywords: ﬁnite diﬀerence, complex-valued function, approximation, error.

2010 Mathematics Subject Classification: 30E05, 30E10 References: [1] N.I. Muskhelishvili, Singular Integral Equations, Noordhoff International Publishing, Groningen, 1953. [2] I.N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford, 1962. [3] V.N. Monakhov, Boundary-Value Problems with Free Boundaries for Elliptic Systems of Equations (Translations of Mathematical Monographs), AMS, 1983. [4] F.D. Gakhov, Boundary Value Problems, Courier Dover Publications, 1990. [5] C. Ashyralyev, Numerical algorithms of the solution for singular integral equations and their applications in hydrodynamics, Ylym, Ashgabat, 1994. [6] A.A. Samarskii, The theory of difference schemes, Marcell Dekker, Inc., New York, USA, 2001. [7] R. Abreu, D.Stich, J. Morales, On the generalization of the Complex Step Method, Journal of Computational and Applied Mathematics, vol. 241, 84–102, 2013. [8] R. Abreu, Complex Steps Finite Differences with applications to seismic problems, PhD Tesis, Universidad de Granada, Granada, Spain, 2013.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 34

Some Parseval-Goldstein type identities for the generalized Macdonald and Hankel transforms

Ne¸seDerneka, Ahmet Derneka, Osman Yureklib a. Department of Mathematics, Marmara University, TR-34722 Kadık¨oy, Istanbul, Turkey. b. Department of Mathematics, Ithaca College, Ithaca, NY-14850, USA. Abstract

In the present paper the authors show that iterations of the generalized Hankel transform with the generalized Macdonald transform is a constant multiple of the generalized Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems of these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained. The identities proven in this paper are shown to give rise to useful corollaries for evaluating inﬁnite integrals of special functions. Some examples are also given as illustration of the results presented here.

Keywords: Hankel transforms, Macdonald transforms, Widder transforms, Laplace transforms, Parseval-Goldstein type identities, Goldstein type exchange identities. 2010 AMS Mathematics Subject Classiﬁcation: Primary 44A10, 44A15, secondary 33C10, 44A35

References:

[1] A. Dernek, et al; Identities for the Hankel transform and their applications; Journal of Mathe- matical Analysis and Applications, (2009), 354; 165- 176. [2] A. N. Dernek, F. Aylık¸cı;Some results on the Pν,2n , Kν,n and Hν,n− integral transforms, Turkish Journal of Mathematics, (2017), 41; 337- 349. [3] A. N. Dernek, F. Aylık¸cı;Identities for the Ln− transform, the L2n− transform and the P2n− transform and their applications. Journal of Inequalities and Special Functions, (2014), 5: 1-16. [4] A. N. Dernek, et al; A generalization of the Widder potential transform and applications. Integral Transform and Special Functions (2011), 22; 391- 401. [5] A. N. Dernek, et al; Parseval-Goldstein type identities involving the L4− transform and the P4− transform and their applications. Integral Transform and Special Functions (2007), 18: 6; 397- 408. [6] A. Erdˇelyi,W., Magnus, F., Obberhettinger, and F. G. Tricomi, Tables of Integral Transforms, Vol. I, Mc. Graw Hill Book Company, New York (1954). [7] A. Erdˇelyi,W., Magnus, F., Obberhettinger, and F. G. Tricomi, Tables of Integral Transforms, Vol. II, Mc. Graw Hill Book Company, New York (1954). [8] I. S. Grandshteyn, I. M. Ryzhik; Table of Integrals, Series and Products, sixthed, Academic Press, (2000). [9] I. J. Hirschman, D. V. Widder; The Convolution Transform, Princeton University Press, Prince- ton, NJ, (1955). [10] S. Kocaba¸set al; Some results on the generalized Mellin transforms and applications; ICAA. (2016) 2nd International Conference on Analysis and Applications July 12- 15, 2016, Turkey. [11] O., Yürekli,A Parseval- type theorem applied to certain integral transforms, Journal of Applied Mathematics, 42: 241- 249, (1989). [12] O., Yürekli, Identities, inequalities, Parseval type relations for integral transforms and fractional integrals. ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)-University of California, Santa Barbara. [13] H. M. Srivastava, O. Yürekli;A theorem on a Stieltjes- type integral transform and its applications, Complex Var. Theory Appl., 28: 159- 168 (1995). [14] J. Spanier, K. B. Oldham; An Atlas of Functions, Hemisphere Publishing Corporation, New York (1954). [15] D.V. Widder, A transform related to the Poisson integral for a half- plane. Duke Mathematica J. 33: 355- 362, (1966). [16] D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, (1971).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 35

A study on Heron triangles and diﬀerence equations Abdullah Sel¸cukKURBANLI Selcuk University, Faculty of Engıneerıng, Department of Metallurgical and Materials Engineering, 42130 Konya, Turkey, [email protected]

Abstract: In this paper, we investigate the relationship between Heron triangles 3 formed by consecutive ordered triple (un − v, un, un + v) ∈ Q and Difference Equation where u, n, v are rational numbers. Keywords: Heron Triangle, Difference Equation, Consecutive Trios. Linear equations with two variables root of real numbers indicates a line. However find the integer roots of same equation requires a completely different approach. Similar to obtain the length of the side and the area which are triangles with rational of the family of triangles requires a different method. This paper produces an infinite family of triangles with rational sides and rational areas and, in fact, an infinite family of triangles with integer sides and integer areas. The results are quite attractive and open a door to a further study of these triangles, both geometrically and number-theoretically. In [1], Buchholz and Rathbun (1997) studied an infinite set of Heron triangles with two rational medians. In [2], Buchholz and Rathbun (1998) presented a proof that there exist infinitely many rational sided triangles with two rational medians and rational area. These triangles correspond to rational points on an elliptic curve of rank one. They also displayed three triangles (one previously unpublished), which do not belong to any of the known infinite families. In [3], Sastry applied a technique in his study that uses a triple of integers that can represent the lengths of the sides of a triangle to generate a family of Heron triangles. In [7], Chisholm and MacDougall examined the remaining six configurations, which were left unsolved by Buchholz [R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1991) 353–368] from the fifteen configurations for tetrahedra having integer edges and volume, by restricting attention to those with two or three different edge lengths, for integer volume, completely solving all but one of them. In [9], Ionascu et al. studied the function H(a, b), which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e., it has integral area. In particular, they proved that H(p, q) ≤ 5 if p and q are primes, and that H(a, b) = 0 for a random choice of positive integers a and b. In this study, it is investigated the n which is rational numbers, so that obtained consecutive triples that makes a triples of Heron against each value given to v in 3 (un(k) − v, un(k), un(k) + v) ∈ Q which and u, v are rational numbers, can be ob- 1 p 2 2 tained from the equation n(k) = u 3m (k) + 4 , where the where m(k) is derived √ k √ k from m(k) = √1 v 2 + 3 − 2 − 3 , k = 0, 1, 2,..., which are the solutions 3 of the linear difference equation m(k + 2) − 4m(k + 1) + m(k) = 0 with the initial conditions m(0) = 0 and m(1) = 2v.

Theorem 2.1. The Heronian triangles in the form (un − 1, un, un + 1) are obtained from diﬀerence equation m(k + 2) − 4m(k + 1) + m(k) = 0 , initial conditions of which 2 4 are m(0) = u and m(1) = u . The area of such triangles are obtained from the formula n √ 2k √ 2ko (2+ 3) −(2− 3) where k = 0, 1, 2,.... 4k = 4

Theorem 2.2. The Heronian triangles in the form (un − 2, un, un + 2) are obtained from diﬀerence equation m(k + 2) − 4m(k + 1) + m(k) = 0 , initial conditions of which 4 8 are m(0) = u and m(1) = u . The area of such triangles are obtained from the formula √ √ 2k √ 2k 4k = 3 2 + 3 − 2 − 3 where k = 0, 1, 2,.... Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 36

Theorem 2.3. The Heronian triangles in the form (un − v, un, un + v) are obtained from diﬀerence equation m(k + 2) − 4m(k + 1) + m(k) = 0 , initial conditions of which are m(0) = 0 and m(1) = 2v . The area of such triangles are obtained from the formula √ √ 2k √ 2k 4k = 3 2 + 3 − 2 − 3 where k = 0, 1, 2,...

References: [1] R.H. Buchholz and R.L. Rathbun. An inﬁnite set of heron triangles with two rational medians. Am. Math. Mon., 104(2), pp 107– 115, 1997. [2] R.H. Buchholz and R.L. Rathbun. Heron triangles and elliptic curves. Bull. Aust. Math. Soc., 58(3), pp 411–421, 1998 [3] K. R. S. Sastry, ”Heron Triangles: A New Perspective”, Australian Mathematical Society Gazette, Vol. 26, Number 4, pp 161-168, 1999 [4] A. Gurbanliyev ve H. S¸enay, “Heron Ucgenleri Uzerine”, Journal of Qafqaz Univercity, 10, pp 227-231, 2004. [5] S. Stevic, More on a rational recurrence relation, Applied Mathematics E-Notes, 4, pp 80-84, 2004. [6] K. R. S. Sastry, “Construction of Brahmagupta n-gons”, Forum Geometricorum, Volume 5, pp 119–126, 2005. [7] C. Chisholm and J.A. MacDougall. Rational and heron tetrahedra. J. Number Theory, 121(1), pp 153–185, 2006. [8] W. H. Richardson, “Super Heron Triangles”, http://www.math.wichita.edu, 2007 [9] E.J. Ionascu, F. Luca, and P. Stianicia. Heron triangles with two ﬁxed sides. J. Number Theory, 126(1), pp 52–67, 2007. [10] S. Kurz, “On The Generatıon Of Heronıan Trıangles”, Serdica J. Computing 2 pp 181-196, 2008

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 37

An inverse problem of heat conduction in a degenerating domain Muvasharkhan Jenaliyev1, Madi Yergaliyev2 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected] 2 Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected]

Abstract: This report is devoted to an inverse problem of ﬁnding a source parameter λ(t) and u(x, t) in following heat equation:

(1) ut(x, t) = uxx(x, t) − λ(t)u(x, t), 0 < x < t, t > 0, with homogeneous initial condition (2) u(x, 0) = 0, and the boundary condition

(3) u(x, t)|x=0 = 0, u(x, t)|x=t = 0, subject to the overspeciﬁcation Z t (4) u(x, t)dx = E(t),E(0) = 0, 0 where the function E(t) is given. In this paper we found a nontrivial solution of the inverse problem for the heat equation in a degenerate domain that satisﬁes the integral condition (4) and found the lambda parameter. It was also shown that the found nontrivial solution is a bounded function for ∀t > 0. Throughout this note we mainly use techniques from our works [1], [2].

Keywords: An inverse problem, heat equation, degenerating domain, integral condition.

2010 Mathematics Subject Classiﬁcation: 35J05, 35J08, 35J25 References: [1] M.M. Amangalieva, M.T. Jenaliyev, M.T. Kosmakova, M.I. Ramazanov, On the boundary value problem of heat conduction with a free boundary, Nonclassical equations of mathematical physics, vol. 2012, 29-44, 2012. [2] M.M. Amangalieva, M.T. Jenaliyev, M.T. Kosmakova, M.I. Ramazanov, On a Volterra equation of the second kind with ”incompressible” kernel, Advances in Diﬀerence equations, vol. 2015, number 71, 1-14, 2015.

The Solution of One Dimensional Fractional Hyperbolic Partial Diﬀerential Equations Using Sumudu Transform Fadime Dal 1 1 Department of Mathematics, Ege University, Izmir, Turkey [email protected]

Abstract: In this study, we propose a new algorithm to ﬁnd exact solution of one dimensional hyperbolic partial diferential equations.The Sumudu transform method can be used to get exact solutions of one dimensional fractional hyperbolic partial diﬀerential equations.The comparison of variational iteration method and these method is presented. Key Words: Fractional hyperbolic equation; initial boundary value problems; sumudu transform method, iteration method

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 38

The Gauss Decomposition for the Group SL(2, R) and Casimir Operator Yasemin Isik1, Mehmet Sezgin2 1 Trakya University, Department of Mathematics, Edirne, Turkey [email protected] 2 Trakya University, Department of Mathematics, Edirne, Turkey [email protected]

Abstract: We considered the Gauss decomposition for the Lie group SL(2, R) . The Casimir operator of the group and continuous state solution of the system were obtained. The Laplace-Beltrami operator on the surface of the manifold was given and the relation between the Casimir operator and the Laplace-Beltrami operator was observed.

Keywords: Lie group, Gauss decomposition, Casimir operator

2010 Mathematics Subject Classification: 22Exx, 57S25, 22E70 References: [1] J. G. F. Belinfante and B. Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods (SIAM, USA, 1972). [2] A. O. Barut and R. Raczka, Theory of Group Representations and Applications (World Scientific Publishing, Singapore, 1972). [3] N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions, Vol I (Kluwer Academic Publishers, The Netherlands, 1991). [4] H. Casimir and B. L. Van der Waerden, Mathematische Annalen, 111, 1-2, (2004). [5] G. Racah, Group Theory and Spectroscopy (Lecture Notes Institute for Advanced Study, Princeton, 1951). [6] A. M. Perelomov and V. S. Popov, Izv. Akad. Nauk SSSR, Vol 2, 6, pp. 1313-1335, (1968). [7] F. Iachello, Lie Algebras and Applications (Springer, Berlin, 2006). [8] A. Jeffrey and H. H. Dai, Handbook of Mathematical Formulas and Integrals (Elsevier, USA, 2008). [9] S. A. Baran, A. Dalgic and G. A. Kerimov, J. Math. Phys. Vol 42, 2, pp. 543, (2001). [10] S. Helgason, Groups and Geometric Analysis (Academic Press, USA, 1984). [11] R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications (John Wiley & Sons, USA, 1974). [12] B. C. Hall, Lie Groups, Lie Algebras and Representations (Springer, Switzerland, 2015). [13] M. Sezgin, A. Y. Verdiyev and Y. A. Verdiyev, Hadronic Journal, 22, pp. 135-144, (1999).

Investigation of the utility function in the problem of consumer choice Issin Meyram 1 1 Faculty of mechanics and mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan [email protected]

Abstract: Interesting and useful for studying is the general model of consumer choice with an arbitrary number of products and a general objective objective function. References: [1] Isin M. Ye., Karpets I. Application of a function of several variables in solving the extreme problem of economics. Materials of the International Scientiﬁc Conference for Young Scientists, Students and Schoolchildren ”VII Satpaev Readings”., Pavlodar, 2007. - t. 18. - P. 154-160. [2] Zamkov OO, Tolstopyatenko AV, Cheremnykh Yu. N. Mathematical Methods in Economics., Moscow State University. MV Lomonosov, DIS, 1997. - 368 p.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 39

Travelling wave solutions for the two dimensional Hirota system of equations Guldana Bekova 1, Gaukhar Shaikhova 1, Kuralay Yesmakhanova 1 1 Department of General and Theoretical Physics, Eurasian National University, Astana, Kazakhstan E-mail: [email protected]

Abstract: In this talk, we investigate the two dimensional Hirota system of equations by next form

(1) iqt + αqxy + iβqxxy − vq + i(wq)x = 0, 2 ∗ ∗ (2) vx + 2αδ(|q| )y − 2iβδ(qxyq − q qxy) = 0, 2 (3) wx − 2βδ(|q| )y = 0, where q(x, y, t) is complex function, v(x, y, t), w(x, y, t) are real functions, α, β, δ are constants. The equations (1)-(3) were presented in work [1], [2]. We apply the extended tanh method for the two dimensional Hirota system of equations in order to obtain new exact solutions. The extended tanh method constructs traveling wave solutions in terms of a hyperbolic tangent functions [3]. New families of solitary wave solutions and periodic solutions are obtained for the two dimensional Hirota system of equations.

Keywords: Hirota system, extended tanh method, traveling wave solution

2010 Mathematics Subject Classiﬁcation: 35C07, 35Q51 References: [1] R. Myrzakulov , G. K. Mamyrbekova , G.N. Nugmanova and M. Lakshmanan Integrable (2+ 1)-dimensional spin models with self-consistent potentials, Symmetry, Vol., 7(3), pp.1352-1375, 2015. [2] K. Yesmakhanova , G. Shaikhova , G. Bekova , Soliton solutions of the Hirota system AIP Con- ference Proceedings 1759, P. 020147 (1-5), 2016. [3] A. Wazwaz, Partial diﬀerential equations and solitary waves theory (Publisher: Springer, Berlin, 2009), pp. 491-493.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 40

Asymptotic expansion of the solution of the nonlocal boundary value problem with initial jumps for singularly perturbed integro-diﬀerential equation Aziza Mirzakulova1, Muratkhan Dauylbaev2 1 Abay Kazakh National Pedagogical University, Kazakhstan [email protected] 2 al-Farabi Kazakh National University, Kazakhstan and Institute of Mathematics and Mathematical Modeling, Kazakhstan, [email protected]

Abstract: This report is devoted to the nonlocal boundary value problem with initial jumps for the integro-diﬀerential equation with a small parameters in the highest derivatives [1]: 1 Z 1 2 000 00 0 X (i) (1) Lεy ≡ ε y + εA0(t)y + A1(t)y + A2(t)y = F (t) + Hi(t, x)y (x, ε)dx, 0 i=0 with the boundary conditions 1 Z 1 0 X (i) (2) y(0, ε) = α, y (0, ε) = β, y(1, ε) = γ + ai(x)y (x, ε)dx, 0 i=0 Here ε > 0 is a small parameter, α, β, γ are known constants.

In this paper we consider the case when the roots µi(t), i = 1, 2 of the additional 2 characteristic equation µ (t) + A0(t)µ(t) + A1(t) = 0 have opposite signs. Asymptotic expansion of the solution of the nonlocal boundary value problem with initial jumps for linear singularly perturbed third-order integro-diﬀerential equations is constructed with any degree of accuracy with respect to a small parameter. The authors were supported in parts by the MESRK grant A` G05132587˘ ”Boundary value problems for singularly perturbed diﬀerential equations with a continuous and piecewise constant argument” (2018-2020) of the Committee of Science, Ministry of Education and Science of the Republic of Kazakhstan.

Keywords: singular perturbations, small parameter, asymptotic expansion

2010 Mathematics Subject Classiﬁcation: 34B05, 34B10, 34E05, 34E10 References: [1] M.K. Dauylbaev, A.E. Mirzakulova, Asymptotic behavior of solutions of singular integro- diﬀerential equations, An Interdisciplinary Journal of Discontinuity, Nonlinearity and Complexity, vol. 5, no 2, 147–154, 2016.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 41

Vertex magic total labeling of selected trees Murat Tezer Department of Mathematics Education, Ataturk Education Faculty Near East University, Northern Cyprus, 99138 [email protected] Abstract: A VMT labeling of a graph with v vertices and e edges is deﬁned as a one-to-one map taking the vertices and edges onto the integers 1,2,...,v+e with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex. In MacDougall et. al. [1] along with many interesting results on VMT graphs, the authors observed that, for the given graph a VMT labeling exists if there is not much variation among the degrees of the vertices. In this study further VMT labelings of a class of simple trees with vertex degrees less than four will be given.

Keywords: Vertex magic, total labeling, trees, graphs

2010 Mathematics Subject Classiﬁcation: 05C78, 05C07 References: [1] J. A. MacDougall, M. Miller & W.D. Wallis, Vertex-magic total labelings of graphs. Utilitas Mathematics, 61, 3-21, 2002.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 42

Stability of basis property of a type of problems with nonlocal perturbation of boundary conditions Bayandy Bostanov1, E.S. Temirbekov2 Matin Dauren1 1 Faculty of mechanics and mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan 2 U.A.Dzholdasbekov IMMash, Almaty, Kazakhstan bostanov [email protected], [email protected], [email protected]

Abstract: We consider the problem of determining the position of the conjugate points of a combined trajectory, ensuring the conditions of tangency, continuity, and equality of the radius of curvature. When the arcs of a circular and elliptical trajectory are conjugated at the junction where there is a common tangent, a jump in the centrifugal force occurs due to the inequality of the radii of curvature, which will lead to a shock / 1-2 /. In order to avoid a jump in curvature, the method of inserting a second-order smoothing curve (the transition cone) is applied. There are known methods of continuous, smooth connection of trajectories, which include contours, splines. With such conjuga- tions, the conditions of tangency and continuity of two arcs of curves are satisfied, but the equality condition for the radii of curvature of these curves at the junction point is not satisfied. A mathematical condition for unstressed conjugation of trajectories was defined. Setting the starting point of the connection and using the condition of smoothness (second-order smoothness), one can determine the end point of the transition curve. The process is proposed to be modeled by a rocking mechanism, which allows to visually, quickly and effectively determine the position of the finish point on the circle. Using the method of determining the position of the conjugate points based on the kinematics of the rocking mechanism, it is possible to smoothly join conical arcs satisfying the conditions of continuity, tangency and equality of curvature and to create on their basis new models of treadmills (trajectories) from conical arcs that allow eliminating unwanted impact effects. 2010 Mathematics Subject Classification: 35J05, 35J08, 35J25

References: [1] Temirbekov E.S., Bostanov B.O.: Analiticheskoye opredeleniye plavnogo perekhoda konturov de- taley odezhdy. Izvestiya vysshikh uchebnykh zavedeniy. Technology of the textile industry. 365, 160–165 (2016) [2] Bostanov, B.O., Temirbekov, E.S., Dudkin, M.V.: Planetarnyy vibrovozbuditel’ s ellipticheskoy dorozhkoy. Proceedings of the All-Russian Scientiﬁc and Technical Conference ”The Role of Me- chanics in Creating Eﬀective Materials, Constructions and Machines of the 21st Century”, pp.66– 70. SibADI, Omsk (2006) [3] Shpigelman, M.: Ellipsy, paraboly i giperboly v sovmeshchennykh polyarno-dekarto-vykh koordi- natakh., M., 2006.-460 p. [4] Fox A.,Pratt M.: Vychislitel’naya geometriya, Application in design and production: Per. with the English .- M .: Mir, 1982.-304 p.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 43

On the stability estimation of ﬁnite-diﬀerence analogue of the integral geometry problem with a weight function Galitdin B. Bakanov Akhmet Yassawi University, Kazakhstan [email protected]

Abstract: The problems of integral geometry are to find the functions, which are determined on certain variety, through its integrals on certain set of subvarities with lower dimension. Additionally, the problems of integral geometry are correlated with various solutions (data interpretation objectives of exploration seismology, electro-exploration, acoustics, and inverse problems of kinetic equations, widely used in plasma physics and astro- physics). In recent years, the studies on problems of integral geometry have critical sig- nificance for tomography, which is intensively developing scientific - technic pillar that has several applications in medicine and industry. Therefore, development of various solution methods for the integral geometry problems is actual issue. One of the stimuli for studying such problems is their connection with multidimensional inverse problems for differential equations [1]. In some inverse problems for hyperbolic equations were shown to reduce to integral geometry problems and, in particular, a problem of integral geometry was considered in the case of shift-invariant curves. Mukhometov [2] showed the uniqueness and estimated the stability of the solution of a two-dimensional integral geometry on the whole. His results were mainly based on the reduction of the two-dimensional integral geometry problem. In this work finite - difference analogue of the two-dimensional problem of integral geometry with a weight function are studied. The stability estimate for the considered problem are obtained.

Keywords: integral geometry problem, ﬁnite - diﬀerence problem, solution, stability.

2010 Mathematics Subject Classiﬁcation: 65N06, 65N20, 65N21 References: [1] M.M. Lavrent’ev , V.G. Romanov, Three linearized inverse problems for hyperbolic equations, Soviet Math. Dokl., 7 , (1966), 1650–1652. [2] R.G. Mukhometov, On a problem of integral geometry, Mathematical Problems of Geophysics, 6 , (1975), 212–242 (in Russian).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 44

Six Point Implicit Methods for the Pure Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation S.C. Buranay1, L.A. Farinola1 1 Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus, Via Mersin 10, Turkey [email protected],[email protected]

Abstract: We construct six point implicit difference boundary value problem for the first derivative of the solution u (x, t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Furthermore, for the second order pure derivatives of u (x, t) special six point implicit difference boundary value problems are proposed. A uniform approximation of the order O h2 + τ 2( second order accurate in the spatial varıable x and second order accurate in time t) where h is the step size in spatial variable x and τ is the step size in time is achieved. It is assumed that the initial function belongs to the Hölderspace C10+α, 0 < α < 1, the heat source α α 8+α,4+ 2 5+ function is from the Hölderspace Cx,t , the boundary functions are from C 2 , and between the initial and the boundary functions the conjugation conditions of orders q = 0, 1, 2, 3, 4, 5 are satisfied. Theoretical results are justified by numerical examples.

Peculiarities of Constructing an Optimal Vector Distributed Control with Nonlinear Optimization of OscillationProcesses Elmira Abdyldaeva1, Akylbek Kerimbekov2, ZarinaKabaeva3 1 Kyrgyz-Turkish Manas University, Kyrgyzstan, efa [email protected] 2 Kyrgyz-Russian Slavic University, Kyrgyzstan 3 Kyrgyz National University, Kyrgyzstan

Abstract: Nonlinear optimization problem is investigated for oscillation processes described by Fredholm integro-di¨ıﬀerential equations in partial derivatives when the function of the external source nonlinearly depends on vector distributed control. It is established that, the optimal control procedure is greatly simpliﬁed with vector control. Algorithm is developed for constructing a complete solution of the nonlinear optimization problem.

Keywords: Boundary value problem, generalized solution, vector distributed control, functional, property of equal relations, system of nonlinear integral equations

2010 Mathematics Subject Classiﬁcation: 49L20, 35K20 References: [1] A. Kerimbekov, E. Abdyldaeva, U. Duishenalieva, E.Seidakmat kyzy, On solvability of optimization problem for elastic oscillations with multipoint sources of control // International Conference ”Functional analysis in interdisciplinary applications” (FAIA2017), AIP Conference Proceedings 1880, edited by Tynysbek Kal’menov and Makhmud Sadybekov (American Institute of Physics, Melville, NY, 2017), 060009 (2017)

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 45

Processing of big data in the detection of geochemical anomalies of rare-earth metal deposits Temirbekova Laura 1 1 Abai Kazakh National Pedagogical University, Kazakhstan [email protected]

Abstract: The paper presents a numerical method for processing a large amount of data in detection of anomalies of the chemical elements distribution in polymetallic deposits. Mathematically, the problem reduces to solving the Fredholm integral equation of the first kind [1-3] for large number of different right-hand parts, while the kernel remains unchanged. b Z (1) K(x, s)u(s)ds = f(x), x ∈ [a, b]. a The algorithm consists of two stages: at the first stage a number of problems are solved, for which knowing the right part is not required, in the second stage, the previously obtained data are used to find solutions of the integral equation for different right-hand sides. This approach allows the data processing ”in-situ” and to define prospects of the established abnormal areas, allocate the overriding areas for geological exploration. The algorithm is tested on model and real data.

Keywords: ill-posed problem, inverse problem, integral Fredholm equation of the ﬁrst kind, kernel, conjugate equation, Fourier series, Fourier coeﬃcients, regularization Lavrentieva, geochemical parameters, chemical elements

2010 Mathematics Subject Classification: 45B05, 86A22, 49N45, 65Q05 References: [1] Karchevsky A.L. Reformulation of an inverse problem statement that reduces computational costs // Eurasian Journal of Mathematical and Computer Applications, 2013, v. 1, n. 2, p. 4-20. https://drive.google.com/file/d/0B2IhUwjYFHEBOUFSTVVfeHhlWFk/view [2] Karchevsky A.L., Marchuk I.V., Kabov O.A. Calculation of the heat flux near the liquid-gas- solid contact line // Applied Mathematical Modelling, 2016, v. 40, n. 2, p. 10291037. doi: 10.1016/j.apm.2015.06.018 [3] Temirbekova L.N., Dairbaeva G. Gradient and Direct Method of solving Gelfand-Levitan Integral Equation // Applied and Computational mathematics, 2013, v. 12, n. 2, p. 234-246.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 46

Iterative Method for Solving Non-Linear Navier-Stokes Equations in Complex Domains Taking Into Account Boundary Conditions with Uniform Accuracy Nurlan Temirbekov1, Zhadra Zhaksylykova2 1 Kazakhstan Engineering Technological University, Almaty, Kazakhstan [email protected] 2 Abay Kazakh National Pedagogical University, Almaty, Kazakhstan [email protected]

Abstract: In this report the Navier-Stokes equations describing the motion of vis- cous incompressible fluid in a bounded domain is considered. Method of fictitious domains is applied for approximate solution of the problem taking into account boundary conditions with uniform accuracy. The original problem is reduced to solving an auxiliary problem corresponding to the method of fictitious domains. A theorem on the existence of the generalized solution to the auxiliary problem is proved. In addition, the convergence of the solution of the auxiliary problem to the solution of the original problem is proved. Next, a difference scheme of the second order of approximation is constructed for the auxiliary problem. For a numerical solution of this difference problem, a special iterative method is constructed that determines approximate solutions on the boundary with uniform accuracy for a limited number of arithmetic operations. To develop a new numerical implementation algorithm, the idea of the fictitious unknowns method with a two-step iterative process and a method for solving the Poisson difference equation in a square with the right-hand side different from zero only at nodes that are a distance of the order of the grid distance from a given piecewise smooth curve are used.

Keywords: method of ﬁctitious domains, Navier-Stokes equations, numerical solution, auxiliary problem, convergence.

2010 Mathematics Subject Classiﬁcation: 65M85, 65M06, 65N22

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 47

A third order of accuracy diﬀerence scheme for Bitsadze-Samarskii type multi-point overdetermined elliptic problem Charyyar Ashyralyyev 1,2, Gulzıpa Aky¨uz 1,3 1Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey

E-mail: [email protected], [email protected]

Abstract: Let us ki, λi, 1 ≤ i ≤ q; λ0, T, α are known nonnegative real numbers such that q P ki = 1, ki ≥ 0, i = 1, . . . , q, (1) i=1 0 < λ1 < λ2 < . . . < λq < T, λ0 ∈ (0,T ) , 0 < α < 1.

Consider the following multipoint multidimensional elliptic problem with overdetermination  n P  −vtt(t, x) − (ar(x)vxr (t, x))xr + σv(x) = g(t, x) + p(x),  r=1   x = (x1, ...xn) ∈ Ω, 0 < t < T, (2) q P  v(0, x) = φ(x), v(T, x) − kiv(λi, x) = η(x),   i=1  v(λ0, x) = ζ(x), x ∈ Ω, v(t, x) = 0, x ∈ S, n where Ω = (0, `) is the open cube in Rn with boundary S, Ω = Ω ∪ S and nonnegative real numbers σ, λ0, λi, ki, 1 ≤ i ≤ q are known, smooth functions ar, φ, η, ζ, and f are given on Ω, ar(x) > 0, ∀x ∈ Ω. In [1], well posedness of problem 2 was established. A first and a second order difference schemes for its approximately solution were constructed. In [2], overdetermined problem for the multi-dimensional elliptic equation with Neumann boundary condition was investigated. The papers [3–5] are devoted to high order approximations of overdetermined elliptic problems without nonlocal boundary conditions. In the present work, we construct a third order of accuracy difference scheme for problem 2 and establish stability, almost coercive stability and coercive stability estimates for its solution. Keywords: nonlocal boundary condition,inverse elliptic problem, well-posedness, stability, coercive stability, overdetermination. 2010 Mathematics Subject Classification: 35N25, 35J67 References: [1] C. Ashyralyyev, G. Akyuz, Stability estimates for solution of Bitsadze-Samarskii type inverse elliptic problem with Dirichlet conditions, AIP Conference Proceedings, vo. 1759 (020129), 2016. [2] C. Ashyralyyev, G. Akyuz, M. Dedeturk, Approximate solution for an inverse problem of multidimensional elliptic equation with multipoint nonlocal and Neumann boundary conditions, Elec- tronic Jourlal of Differential Equations, vol. 2017, no 197, 1-16, 2017. [3] C. Ashyralyyev, High order approximation of the inverse elliptic problem with Dirichlet-Neumann Conditions, Filomat, vol. 28, no 5, 947–962, 2014. [4] C. Ashyralyyev, High order of accuracy difference schemes for the inverse elliptic problem with Dirichlet condition, Boundary Value Problems, vol. 2014, no 5, 1–23, 2014. [5] C. Ashyralyyev, A fourth order approximation of the Neumann type overdetermined elliptic problem, Filomat, vol. 31, no 4, 967-980, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 48

Multi-Objective mean-CVaR model under VG process on evaluating Portfolio eﬃciency

Shokoofeh Banihashemi 1, Masoumeh Mirsadeghpour 2, Navideh Modarresi 1 1 Department of Mathematics, Faculty of mathematics and Computer Science, Allameh Tabataba’i University, Tehran, Iran E-mail: [email protected] 2 Department of Applied Mathematics,Central Tehran Branch, Islamic Azad University E-mail: [email protected]

Abstract:In this study, we evaluate the portfolio performance by Data Envelopment Analysis (DEA) as a nonparametric eﬃciency analysis tool. Our model describes the dynamics of assetsˆa2122 log prices by a stochastic process which is named Variance- Gamma (VG) process [1]. Risk measure of our model is Conditional Value at Risk (CVaR)[2]. Therefore, the model is in mean-CVaR framework under VG process. Con- ventional DEA models consider mainly nonnegative data. However, in real data world, inputs and outputs may as well take negative values. So, our proposed model is the Range Directional Measure-like model that can take positive and negative values[3]. Fi- nally, we present a case study of the stock market to demonstrate the applicability of the proposal model. For estimating the parameters in the model we use Monte Carlo approach and a nonlinear programming technique.

2010 Mathematics Subject Classiﬁcation: 91G10,91G60 References: [1] D. B. Madan and E. Senata, “Option pricing under VG , Journal of business, Vol. 63, pp.511-524, 1990. [2] R. T. Rockafellar and S. Uryasev, “Portfolio Optimizatin with CVaR, Journal of Risk, Vol. 2, pp.21-41, 2000. [3] M. C. Portela and E. Thanassoulis, “A Directional distance approch to deal with negative data in DEA, European Journal of Operational Research, Vol. 10, pp.1111-1121, 2004.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 49

Stationary boundary-value problems of the anisotropic elastic medium dynamics Gulmira Zakiryanova Institute of Mechanics and Mechanical Engineering after academician U.A.Dzholdasbekov, Kazakhstan [email protected]

Abstract: The solution of boundary value problems of dynamics of elastic media is one of the topical problems of mathematical physics, because by them is described a wide class of various geophysical processes, dynamic processes in materials and constructions that are widely used in bilding and engineering. The most are developed the mathematical methods for investigation of boundary value problems of the dynamics of isotropic elastic media. Anisotropic media have been little studied, which is connected with the complexity of constructing solutions of equations that are usually solved on the basis of numerical diﬀerence or ﬁnite element methods. Here we consider the solution of problems of stationary oscillations with a frequency ω for anisotropic elastic media, which for complex amplitudes is described by a system of equations of the form

(1) Lij (∂x, −iω) uj (x) + Gi(x) = 0,

ml 2 (2) Lij (∂x, −iω) = Cij ∂m∂l + δij ρω , i, j, m, l = 1,N

Here ui,j = ∂ui/∂xj , δij is Kronecker symbol, ρ is the density of the medium, Gi are the ml components of the mass force, Cij is the matrix of elastic constants, which has symmetry ml lm ml ij properties with respect to permutation of the indexes Cij = Cij = Cji = Cml (everywhere the summation over repeated indices in the indicated range is assumed). Two boundary value problems for complex amplitudes of displacements of the medium, which belong to the class of boundary value problems for elliptic systems of equations, are given. To solve them we use the method of generalized functions that allows the initial boundary value problem to lead to the solution of a system of differential equations in the space of generalized functions with a certain right-hand side from the class of singular generalized functions of the type of simple and double layers and to use the properties of fundamental solutions of such systems for constructing its generalized solutions.Considered the fundamental solutions, their symmetry properties and asymptotics. A generalization of the Somilyana formula for displacements is obtained which allows by known displacements and stresses on the boundary of body one to find the displacement amplitudes inside it. The analogue of Gauss’s formula is presented, which determines the characteristic function of the domain of definition through surface integrals of the stress tensor. On its basis singular boundary integral equations for solving the boundary value problems are constructed. This work was supported by the Ministry of education and science of Republic of Kazakhstan by Grant No.AP05135494.

Keywords: anisotropic medium, elasticity, fundamental solutions, boundary value problems, stationary oscillations, boundary integral equations

2010 Mathematics Subject Classiﬁcation: 74B05, 74E10

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 50

On the system of rational diﬀerence equations

Abdullah Sel¸cukKurbanli Selcuk University, Faculty of Engıneerıng, Department of Metallurgical and Materials Engineering, 42130 Konya, Turkey, [email protected]

Abstract: In this paper, we study the following system of diﬀerence equations for n ∈ N0

 xn−1 xn+1 = ,  ynxn−1−1  yn−1 (1) yn+1 = x y −1 ,  n n−1  zn+1 = xnynzn−1, where x0, x−1, y0, y−1, z0, z−1 real numbers such that y0x−1 6= 1 and x0y−1 6= 1.

Theorem 2.4. Let y0 = a, y−1 = b, x0 = c, x−1 = d, z0 = e, z−1 = f be real numbers such that ad 6= 1 and cb 6= 1. Let {xn, yn, zn} be a solution of the system (1). Then all solutions of (1) are ( d (ad−1)n , n − odd, (2) xn = c(cb − 1)n, n − even

( b (cb−1)n , n − odd, (3) yn = a(ad − 1)n, n − even

 n−1  Σ i  (ca)nf[(cb − 1) (ad − 1)] i=0 , n − odd, (4) zn = dnbne n , n − even  Σ i  [(ad−1)(cb−1)]i=1

References: [1] A. S. Kurbanli, C. C¸inar, I.˙ Yalcinkaya, Mathematical and Computer Modelling, 53(5-6) 1261- 1267 (2011). [2] A. S. Kurbanli, Advances in Diﬀerence Equations, 2011:40 1-8 (2011). [3] A. S. Kurbanli, Discrete Dynamics in Nature and Society, 2011, Article ID 932362, 12 pages, doi:10.1155/2011/932362 (2011).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 51

Special Functions and Heat Polynomials for the Solution of Free Boundary Problems Stanislav N. Kharin1,2 1 Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 050010 Almaty, Kazakhstan 2 Kazakh- British Technical University, 59, Tole-bi str., 050000, Almaty, Kazakhstan [email protected]

Abstract: The method of special functions and heat polynomials for the solution of problems for the heat equation in domains with moving boundaries is developed for the generalized and axisymmetric heat equations. The corresponding generating and associated functions are introduced. Examples of application of this method for the solution of some problems are presented. This research is ﬁnancially supported by a grant AP05133919 and by the target program BR05236656 from the Science Committee from the Ministry of Science and Education of the Republic of Kazakhstan. References: [1] P.C. Rosenbloom and D.V. Widder, Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc., Vol. 92 (1959), 220-266. [2] S.N.Kharin, M.M.Sarsengeldin, H.Nouri, Analytical solution of two phase spherical Stefan problem by heat polynomials and integral error functions, AIP Conference Proceedings, 1759, 020031 (2016); doi: http://dx.doi.org/10.1063/1.4959645. [3] M. Sarsengeldin, S.N. Kharin, Method of the Integral Error Functions for the Solution of the One- and Two-Phase Stefan Problems and Its Application,Filomat 31:4 (2017), 1017–1029 DOI 10.2298/FIL1704017S, Available at: http://www.pmf.ni.ac.rs/ﬁlomat.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 52

Numerical solution of stochastic mixed Volterra-Fredholm integral equations driven by space-time Brownian motion via two-dimensional block pulse functions Farkhondeh Hosseini Shekarabi1 1 Department of Mathematics, Yadegar-e-Imam Khomeini (Rah) Shahre Rey Branch, Islamic Azad University, Tehran, Iran f [email protected]

Abstract: The goal of this presentation is to give useful method for solving problems formulated by two dimensional stochastic integral equations driven by space-time white noise. Typically, some stochastic problems require numerical methods to obtain a solution and therefore is focused on stochastic integral and partial differential equations to construct a reliable and efficient computational methods. Since solution of parabolic partial differential equations driven by space-time white noise leads to a stochastic integral equations, we introduce a new method by two dimensional block pulse functions and their operational integration matrix to transform a stochastic mixed Volterra-Fredholm integral equation to a system of algebraic equations. The benefit of this method is lower cost of setting up the system of equations without any integration. So, the computational cost of operations is low. The method is applied to test examples to illustrate the accuracy and implementation of the method.

Keywords: Two-dimensional block pulse functions; Operational matrix; Stochastic operational matrix; Stochastic mixed Volterra-Fredholm integral equations; Space-time white noise; Brownian sheet.

2010 Mathematics Subject Classiﬁcation: Primary: 65C30, 60H35, 65C20; Sec- ondary: 60H20, 68U20, 60H15. References: [1] M. Khodabin, K. Maleknejad, F. Hosseini Shekarabi, Application of Triangular Functions to Nu- merical Solution of Stochastic Volterra Integral Equations, IAENG International Journal of Ap- plied Mathematics, 43:1,(2013), IJAM-43-1-01. [2] F. Hosseini Shekarabi, M. Khodabin, Numerical Solution of a Model for Stochastic Polymer Equa- tion Driven by Space-Time Brownian Motion via Homotopy Perturbation Method, International Journal of Applied and Computational Mathematics 06/2015; DOI: 10.1007/s40819-015-0072-4, 2015.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 53

The construction of a solution of a related system of the Laguerre type Zh.N. Tasmambetov1, R.U. Zhakhina 1 1 Department of Computer science and information technology, Aktobe Regional State University named after K. Zhubanov [email protected], riscul [email protected]

Abstract: The systems of the Laguerre type, which were obtained from the system of Horn by direct selection of parameters as well as by exponential transformation, are considered. For the construction of their normal-regular solution, which is depended on the Laguerre polynomial of two variables, the Frobenius- Latysheva method is applied. Classiﬁcation of special curves by means of rank and anti rank, and the basic information about the peculiarities of construction of normal-regular solutions of such systems are presented. The main theorem on the existence of four linearly independent particular solutions, which are expressed in terms of the degenerate hypergeometric function of M.R. Humbert in the form of normally-regular series dependent on Laguerre polynomials of two variables, is proved.

Keywords: Related system, the system of the Laguerre type, the system of Horn, normal-regular solution, special curves, rank, antirank References: [1] E.T.Witteker, G.N.Watson, A course of Modern Analysis. Part 2: Transcendental functions., Cam- bridge. At the university Press, 1927. - 515 pp. [2] G.Jeffries, B.Swirls, Methods of mathematical physics, Issue 3, M.:”Mir”, 1970. - 344 p. [3] G.Beytmen, A.Erdeli, Higher transcendental functions. Part I. Hypergeometric of the functions. The functions of Lezhanders, M.: Science, 1965. - 294 pp. [4] Zh.N.Tasmambetov, Confluent hypergeometric functions and two variables Laguerre polynomials as solutions of Wilczynski type system, AIP Conference Proceedings 1759, 020137 (2016); cloi: 10.1063/1, 4959751. [5] E.J.Wilczynski, Projective differential Geometry of Curves and Ruled Surfaces, Leipzig: Leubner, 1906. - 120 pp. [6] Zh.N.Tasmambetov, Construction of normal and normal-regular solutions of special systems of partial differential equations of the second order, Aktobe: IP Zhanadilova S.T., 2015. - 463 p. [7] Zh.N.Tasmambetov, On irregular singular curves of Witteker type system, Vestnik Sam.GTU, Seria fiziko-mathematisheskie nauki, Issue 4(33), 2013. p.25-33.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 54

The Coinciding Polygonal Numbers and the Related Arithmetical Properties of Tchebyshef Polynomials Yu. Ya. Agranovich Voronezh State Technical University, The Russian Federation [email protected]

Abstract: We present an algorithm for identification of all coinciding heptagonal and square numbers. The solution to the corresponding Diophantine equation is in the form of three series of ascending solutions. We also specify arithmetic progressions containing certain linear combinations of values of Tchebyshef polynomials at odd triangle numbers. This research was supported by the Russian Foundation for Basic Research, RFBR grant 14-01-00253 a. References: [1] Yu. Ya. Agranovich, S. L. Podvalny, I. Y. Novikov, and V. L. Khatskevich, Elements of theoretical computer science, Voronezh State Technical University, Voronezh, 2009. [2] Yu. Ya. Agranovich, N. V. Kontsevaya, and V. L. Khatskevich, Method of polygonal numbers in the procedure of smoothing time series and its application to the research of financial markets parameters, Economics and Mathematical Methods 46(3) (2010), 71–81. [3] Yu. Ya. Agranovich, N. V. Kontsevaya, and V. L. Khatskevich, Smoothing time series of financial markets parameters by method of polygonal numbers, Applied Econometrics (3) (2010), 3–8. [4] N. B. Alfutova and A. V. Ustinov Algebra and number theory, MCCME, Moscow, 2005. [5] S. Ando, On a system of Diophantine equations concerning the polygonal numbers, Fibonacci Quarterly 20(4) (1982), 349–354. [6] D. S. Lucas, Numbers common to two polygonal sequences, Fibonacci Quarterly 11 (1973), 78–84. [7] T. Nagell, Introduction to Number Theory,Almqvist, Stockholm; Wiley, New York, 1951. [8] J. S. Su, On some properties of two simultaneous polygonal sequences, JIS 10 (2007), article 7.10.4, 349–354. [9] N. J. Wildberger, Pell’s equation without irrational numbers, JIS 13(270) (2010), article 10.4.3.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 55

Construction of solutions of inhomogeneous systems of Jacobi type Zhaksylyk N. Tasmambetov1, Meiramgul Zh. Talipova1 1 Department of Computer science and information technology, Aktobe Regional State University named after K. Zhubanov [email protected], mira [email protected]

Abstract: Studying the possibilities of constructing a general solution of a nonhomogeneous system of second-order partial differential equations of Jacobi type, closest to ordinary differential equations of the second order. To construct a particular solution of the system is applied the method of undetermined coefficient. Specific examples are established the connection between systems of Jacobi type with one partial differential equation of the second order obtained by adding two equations of the original system.

Keywords: System, particular solution, homogeneous, inhomogeneous, general solution, polynomial, orthogonal References: [1] Lucy Joan Slater. Generalized hypergeometric function Gambridge at the university press., 1966. 271 pp. [2] Fascicule IX. Fonctions de la physique mathematique. Sous la direction de J.Kampe de Feriet. Centre national de la recherch scientifique., 1957. 102 pp. [3] Yudell L. Luke., Mathematical functions and their approximations Academic Press Inc., New York. San Francisco, London. 1975. 608 pp. [4] Harry Bateman, Arthur Erdelyi. Higher transcendental functions. Volume 2. New York, Toronto, London. MC Graw-Hill, Book Company, INC, 1953, 295 pp. [5] Gabor Szego’. Orthogonal Polynomials. American Mathematical Socity, New York Cit, 1952. 500 pp. [6] Yu.I. Sikorski. Normal solutions of linear inhomogeneous differential equations with variable coefficients, Author’s abstract. ... dis. cand. Kiev, 12 pp. [7] Zh.N.Tasmambetov. Construction of normal and normal-regular solutions of special systems of partial differential equations of the second order, Aktobe: IP Zhanadilova S.T., 2015. - 463 p. [8] Talipova M.Zh. Construction of normal solutions of inhomogeneous systems of partial differential equations of the second order, Author’s abstract. ... dis. cand. Almaty, 2007, 12 pp. [9] Suetin P.K. Orthogonal polynomials in two variables. Science, Moscow.:, 1988, 384 p. [10] Babister A.W. Transcendental Functions Satisfying Nonhomogeneous Linea Differential Equations. Macmil- lan, New York, London, 1967.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 56

Assessment of slope stability (on the example of Jalal-Abad in the area of Khamza street) Mirzaliev M.1, Asilova Z.A.2 1 Institute of Engineering Survey OJSC ”KyrgyzLIES” Osh branch, Kyrgyzstan, e.mail - oshgiiz.rambler.ru 2 Jalal-Abad State University, Kyrgyzstan asilova.zulfi[email protected] Abstract: In this paper, we present the results of an engineering and geological survey to assess the stability of a slope in the area of Khamza street in the city of Jalal-Abad. The climatic characteristics were given from the data of long-term observations at the ”Jalalabad” meteorological station. Geological, hydrogeological, geomorphological and tectonic conditions of the terrain are given. The seismicity of the area is estimated by the initial score 9. The physical and mechanical properties of soils have been determined as a result of the laboratory processing of field and laboratory materials, and in accordance with GOST 20522-75 and GOST 25100-95, two engineering-geological elements (IGE) were identified in the studied area: IGE-1: Loess loam, light brown, mostly solid and semi-solid. Density of loamy soils with depth sustained, deviations insignificant. IGE–2: Conglomerate gray, strong, slightly cracked, from the surface is exposed to weathering. The rock density is: p = 2.2 − 2.4t/m. As a result of the research, the following conclusions were obtained: 1. Currently landslide processes of threatening sizes and volumes on the slope are not observed. The calculations of the coefficient of stability allow us to confirm the stable state of the soil mass of the constituent slope for the period of research. With the value of the coefficient of stability Ku¿ 1, the array is fairly stable. 2. Absence of cracks in subsidence, precursors of possible or expected landslide processes, also points to the stability of the slope at a given time. 3. In the case of precipitation of abnormally abundant precipitation and, as a consequence, wetting of the entire mass of loamy strata, especially in the plantar part, to the soft and flowing-plastic state, deformation processes associated with subsidence of loess-like loams in the upper part of the slope where their thickness is greatest are possible.

Keywords: landslide, stability, engineering-geological conditions, coeﬃcient of stability.

References: [1] Kozhogulov K.Ch., Nikolskaya O.V. Ibatulin H.V. Landslides of the South of Kyrgyzstan. - Bishkek: 1993. -46p.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 57

A Discrete-time Holling Type II Model with Allee and Refuge Eﬀects Unal Ufuktepe1, Burcin Kulahcioglu2 1 American University of the Middle East, Kuwait [email protected] 2 Izmir University of Economics, Department of Mathematics, Turkey [email protected]

Abstract: A discrete-time Holling type II prey-predator model with Allee and the refuge eﬀects is formulated and studied. The existence of the positive equilibrium and Its stability are investigated for both hyperbolic and non hyperbolic cases. Numerical simulations are conducted to demonstrate the theoretical results.

Keywords: Discrete dynamical systems, Holling type

2010 Mathematics Subject Classification: 92B05 References: [1] Hui Cao, Z.Yue, and Y.Zhou, The stability and bifurcation analysis of a discrete Holling-Tanner model, Advances in Difference Equations, 2013:330, (2013) p 1–17. [2] S.B. Hsu, T.W. Hwang, Global stability for a class predator–prey system, SIAM J. Appl. Math. 55 (1995) 763–783. [3] N. Kasarinoff, P. van der Deiesch, A model of predator–prey system with functional response, Math. Biosci. 39 (1978) 124–134. [4] S. Ruan, D. Xiao, Global analysis in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001) 1445–1472. [5] J. Hainzl, Stability and Hopf bifurcation a predator–prey system with several parameters, SIAM J. Appl. Math. 48 (1988) 170–180. [6] H.N. Agiza, E.M. ELabbasy, H. EL-Metwally, A.A. Elsadany, Chaotic dynamics of a discrete prey– predator model with Holling type II,Nonlinear Analysis: Real World Applications 10 (2009) 116–129

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 58

On a ﬁrst-order partial diﬀerential equation Tamasha Aldibekov Al-Farabi Kazakh National University, Kazakhstan [email protected]

Abstract: In this report we study a linear differential equation with first-order partial derivatives, where the coefficients of the equation are given on an unbounded set and has continuous first-order partial derivatives. n ! ∂u X ∂u + p (x)y + g1(x, y1, . . . , yn) ∂x 1k k ∂y k=1 1 n ! X ∂u ... + p (x)y + gn(x, y1, . . . , yn) = 0 (1) nk k ∂y k=1 n where x ∈ I ≡ [x0 ≤ x < +∞), x0 > 0, −∞ < y1, . . . , yn < +∞, Theorem 1. If the following conditions are fullfilled:

(1) pik(x), i = 1, . . . , n, k = 1, . . . , n; are continuous diﬀerentiable on I. (2) pk−1,k−1(x) − pkk(x) ≥ αϕ(x), x ∈ I, k = 2, . . . , n. α > 0, ϕ(x) ∈ C(I), x ϕ(x) > 0, q(x) = R ϕ(s)ds ↑ +∞; x0 (3) lim |pik(x)| = 0, i 6= k, i = 1, 2, . . . , n, k = 1, 2, . . . , n; x→+∞ ϕ(x) x 1 R (4) lim pkk(s)ds = βk, k = 1, 2, . . . , n, β1 < 0; x→+∞ q(x) x0

(5) g(x, y) = colon (g1(x, y1, . . . , yn), . . . , gn(x, y1, . . . , yn)) has a continuous partial derivatives on the set x0 ≤ x < +∞, x0 > 0, −∞ < y1, . . . , yn < +∞, `e gi(x, 0,..., 0) = 0, i = 1, . . . , n. kg(x, y)k ≤ δ(x)kyk, lim δ(x) = 0, then the equation (1) has a integral basis, which attempts x→+∞ ϕ(x) to zero at x0 → +∞.

Keywords: equation, ﬁrst order partial derivatives

2010 Mathematics Subject Classiﬁcation: 35B35 References: [1] V.M.Smirnov, Kurs vyshei matematiki[Higher mathematics course], Moskva:Nauka, 1981, V. 4, 17- 22 pp.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 59

Identiﬁcation hyperbolic problems with nonlocal conditions A. Ashyralyev1, F. Emharab2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Peoples’ Friendship University of Russia (RUDN University), Moscow 117198, Russia. Institute of Mathematics and Mathematical Modeling, 050010, Almaty. [email protected] URL:http://staﬀ.neu.edu.tr 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Department of Mathematics, Omar Al-Mukhtar University, El-Beida. [email protected]

Abstract: In the present study, a identification problem with nonlocal conditions for a one-dimensional hyperbolic equation is investigated. Stability estimates for the solution of the identification problem are established. Furthermore, a first order of accuracy difference scheme for the numerical solution of the identification hyperbolic equations problems with nonlocal conditions is presented. Stability estimates for the solution of the difference scheme are established. Then, this difference scheme is tested on an example and some numerical results are presented. Keywords: Source identification problem, hyperbolic differential equations, difference schemes. 2010 Mathematics Subject Classification: 35J05, 35J08, 35J25

References: [1] G. Di. Blasio, A. Lorenzi, Identification problems for parabolic delay differential equations with measurement on the boundary, Journal of Inverse and Ill-Posed Problems, 15(7):709–734, 2007. [2] A. Ashyralyev, D. Agirseven, On source identification problem for a delay parabolic equation, Nonlinear Analysis: Modelling and Control, 19(3): 335-349, 2014.

[3] A. Ashyralyev, F. Celik, On source identification problem for telegraph differential equations, Differential and Difference Equations with Applications, 164 of the series Springer Proceedings in Mathematics and Statistics, 39-50, 2016.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 60

Numerical Solution of Nonlocal Elliptic Problems Allaberen Ashyralyev1, Ayman Hamad2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Peoples’ Friendship University of Russia (RUDN University), Moscow , Russia. Institute of Mathematics and Mathematical Modeling 050010, Almaty, Kazakhstan. [email protected] 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Department of Mathematics, Omar Al-Mukhtar University, El-Beida, Libya. [email protected]

Abstract: The well-posedness in various Banach spaces of the local boundary value problem for the elliptic equation in an arbitrary Banach space with the positive operator and its related applications have been investigated by many researchers (see, for example, [1]- [3] and the references given therein). In the present paper, the second order of approximation two-step diﬀerence scheme for the approximate solution of the nonlocal boundary value problem for the elliptic differential equation in an arbitrary Banach space with the positive operator is investigated. The well-posedness of this diﬀerence scheme in various Banach spaces is established. In applications, some illustrative numerical results are provided.

Keywords: Well-posedness; coercive stability; positive operators; elliptic equation.

2010 Mathematics Subject Classification: 2010 Mathematics Subject Classifica- tion: 35J25, 47E05, 34B27 References: [1] A. Ashyralyev, P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Op- erator Theory: Advances and Applications, vol.148, Birkhauser, Verlag, Basel, Boston, Berlin 2004. [2] A. L. Skubachevskii, Elliptic functional differential equations and applications, Birkhauser Verlag, Basel, Boston, Berlin, 1997. [3] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhauser Verlag, Basel, Boston, Berlin,1995.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 61

Well-posedness of Neumann-type elliptic overdetermined problem with integral condition Charyyar Ashyralyyev 1,2, Aysel C¸ay1,3 1,2Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey

2TAU, Ashgabat, Turkmenistan E-mail: [email protected], [email protected]

Abstract: In this work, we consider the following Neumann type elliptic overdetermined problem with integral condition to find a function u ∈ C2 ([0,T ] ,H) ∩ C ([0,T ] ,D(A)) and an element p ∈ H :  −u (t) + Au(t) = f(t) + p, 0 < t < T,  tt T R ut(0) = ϕ, ut(T ) = α (λ) uλ(λ)dλ + ψ, u(λ0) = ζ.  0 where A is a selfadjoint and positive definite operator in an arbitrary Hilbert space H, smooth function f(t), the elements ϕ, ζ, ψ ∈ D(A) and number λ0 ∈ (0,T ) are given. T Let the given smooth scalar fuction α (t) be under condition R |α (λ)| dλ < 1. 0 In the papers [1–5] well-posedness of various overdetermined elliptic type differential and difference problems are studied. Abstract results on stability, almost coercive stability and coercive stability estimates for the solution of this problem are established. Later, the abstract results are used to establish well-posedness of overdetermined problem multi-dimensional elliptic equation with integral boundary condition.

Keywords: inverse problem, well-posedness, stability, coercive stability, overdetermination.

2010 Mathematics Subject Classiﬁcation: 35N25, 35J67 References: [1] A. Ashyralyev, C. Ashyralyyev, On the problem of determining the parameter of an elliptic equation in a Banach space, Nonlinear Analysis: Modelling and Control, vol. 19, no 3, 350–366, 2014. [2] C. Ashyralyyev, Stability Estimates for Solution of Neumann Type Overdetermined Elliptic Prob- lem, Numerical Functional Analysis and Optimization vol. 38, no 10, 1226–1243, 2017. [3] C. Ashyralyyev, Numerical Solution to Bitsadze-Samarskii Type Elliptic Overdetermined Multi- point NBVP, Boundary Value Problems, vol. 2017, no 74, 1–22, 2017. [4] C. Ashyralyyev, G. Akyuz, Stability estimates for solution of Bitsadze-Samarskii type inverse elliptic problem with Dirichlet conditions, AIP Conference Proceedings, vo. 1759 (020129), 2016. [5] C. Ashyralyyev, G. Akyuz, M. Dedeturk, Approximate solution for an inverse problem of multidimensional elliptic equation with multipoint nonlocal and Neumann boundary conditions, Elec- tronic Jourlal of Diﬀerential Equations 2017 (197) (2017) 1-16.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 62

On the numerical solution of advection-diﬀusion problems with singular source terms Ezgi Soykan1, Mehmet Ahlatcioglu1,Maksat Ashyraliyev2 1 Department of Mathematics, Yildiz Technical University, Turkey 2 Department of Software Engineering, Bahcesehir University, Turkey [email protected]

Abstract: Partial differential equations with singular source terms are widely used in mathematical modeling of real-life systems in many different fields of science. Singular means that within the spatial domain the source is defined by a Dirac delta function. Solutions of the problems having singular source terms have lack of smoothness, which is generally an obstacle for standard numerical techniques [1–3]. In this work, we consider the initial-boundary value problem with singular source terms  ut + aux = Duxx + k1δ(x − ξ1) + k2δ(x − ξ2), 0 < x < 1, t > 0,   (1) u(t, 0) = uL, u(t, 1) = uR, t ≥ 0,   u(0, x) = ϕ(x), 0 ≤ x ≤ 1, where 0 < ξ1 < ξ2 < 1 and δ(x) is a Dirac delta function. We firstly derive the analytical solution of problem (1). Further, we describe the procedure for numerical solution of problem (1) using the standard finite volume method. We provide with numerical illustration for simple test problem.

Keywords: advection-diﬀusion equation, singular source terms, ﬁnite volume method

2010 Mathematics Subject Classification: 65M08, 65M20, 65N08 References: [1] J.Santos, P. de Oliveira, A converging finite volume scheme for hyperbolic conservation laws with source terms, Journal of Computational and Applied Mathematics, vol. 111, 239–251, 1999. [2] A K. Tornberg, B. Engquist, Numerical approximations of singular source terms in differential equations, Journal of Computational Physics, vol. 200, no 2, 462–488, 2004. [3] M. Ashyraliyev, J.G. Blom, J.G. Verwer, On the numerical solution of diffusion-reaction equations with singular source terms, Journal of Computational and Applied Mathematics, vol. 216, no 1, 20–38, 2008.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 63

A Second Order Difference Scheme for the Numerical Processing of Source Identification Terms in Hyperbolic Equations Allaberen Ashyralyev1, Fathi Emharab2 1Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Peoples’ Friendship University of Russia (RUDN University), Moscow 117198, Russia. Institute of Mathematics and Mathematical Modeling, 050010, Almaty [email protected] URL:http://staff.neu.edu.tr 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Department of Mathematics, Omar Al-Mukhtar University, El-Beida. [email protected]

Abstract: In the present study, a second order difference scheme for a one-dimensional hyperbolic equation is presented. Stability estimates for the solution of the difference scheme are established. This difference scheme is tested on an example and some numerical results are presented. Keywords: Source identification, hyperbolic differential equations, difference schemes, stability. 2010 Mathematics Subject Classification: 35J05, 35J08, 35J25

References: [1] A. Ashyralyev and A. U. Sazaklioglu, Investigation of a time-dependent source identification inverse problem with integral overdetermination, Numerical Functional Analysis and Optimization 38 (10):1276-1294, 2017. [2] M. Ashyralyyeva and M. Ashyraliyev, On a second order of accuracy stable difference scheme for the solution of a source identification problem for hyperbolic-parabolic equations, AIP Conference Proceedings 1759 (2016), 020023. [3] G. Di. Blasio, A. Lorenzi, Identification problems for parabolic delay differential equations with measurement on the boundary, Journal of Inverse and Ill-Posed Problems, 15(7):709-734, 2007.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 64

Riesz basis property of generalized eigenfunctions for many interval BVTP’s with eigenparameter dependent boundary conditions Hayati Olˇgar 1, Oktay Sh. Mukhtarov 1,2 and Kadriye Aydemir3 1 Department of Mathematics, Faculty of Science Art, Gaziosmanpa¸saUniversity, Turkey E-mail: [email protected] 2Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Azerbaijan E-mail: [email protected] 3 Department of Mathematics, Faculty of Science Art, Amasya University, Turkey E-mail: [email protected]

Abstract: The main goal of this study is to provide an operator-pencil framework for the investigation of many-interval boundary-value-transmission problems (BVTP) with eigenparameter appearing in the boundary-transmission conditions. By applying an our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is deﬁned a concept of generalized eigenfunctions. We introduce some self-adjoint compact operators in suitable Sobolev spaces such a way that the considered problem can be reduced to an operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of generalized eigenfunctions form a Riesz basis of the suitable Hilbert space.

Keywords: Boundary value problems, boundary and transmission conditions, eigenvalues, generalized eigenfunctions, Riesz basis.

2010 Mathematics Subject Classiﬁcation: 34L10, 34L15 References: [1] Kadriye Aydemir, Oktay Mukhtarov, A class of sturm-liouville problems with eigenparameter dependent transmission conditions, Numerical Functional Analysis and Optimization, (2017), Doi: 10.1080/01630563.2017.1316995. [2] B. P. Belinskiy and J. P. Daver and Y. Xu, Inverse scattering of acustic waves in an ocean with ice cover, Applicable Analysis 61 (1996), 255-283. [3] I. C. Gohberg and M. G. Krein, “Introduction to The Theory of Linear Non-Selfadjoint Operators”, Translation of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, Rhode Island, 1969. [4] M. V. Keldys, “On the eigenvalues and eigenfunctions of certain classes of non-self-adjoint equations”, Dokl Akad. Nauk SSSR (in Russian) 77 (1951), 11-14; English transl. in this volume. [5] A. G. Kostyuchenko and A.A. Shkalikov, Self-adjoint quadratic operator pencils and eliptic problems, Functional Anal. Appl. 17 (1983), 109-128. [6] O. A. Ladyzhenskaia, “The Boundary Value Problems of Mathematical Physics”, Springer-Verlag, New York, 1985. [7] H. Olˇgarand O. Sh. Mukhtarov, Weak Eigenfunctions Of Two-Interval Sturm-Liouville Problems Together With Interaction Conditions, Journal of Mathematical Physics, 58, 042201 (2017) DOI: 10.1063/1.4979615. [8] J. Walter, “Regular eigenvalue problems with eigenvalue parameter in the boundary conditions”, Math. Z., 133, 301-312, 1973.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 65

Lower bound estimation for eigenvalues for many interval BVP’s with eigenparameter dependent boundary conditions Oktay Sh. Mukhtarov 1,2, Hayati Olˇgar 2 and Kadriye Aydemir3 1Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Azerbaijan E-mail: [email protected] 2 Department of Mathematics, Faculty of Science Art, Gaziosmanpa¸saUniversity, Turkey E-mail: [email protected] 3 Department of Mathematics, Faculty of Science Art, Amasya University, Turkey E-mail: [email protected]

Abstract: The present paper deals with multi-interval Sturm-Liouville equations with eigenparameter dependent boundary-transmission conditions. Such type of problems cannot be treated with the usual techniques within the standard framework os classical Sturmian theory. It is well-known that any eigenvalue of the classical Sturm- Liouville problems can be related to its eigenfunction by the Rayleigh quotient and some useful results can be obtained from the Rayleigh quotient without solving the diﬀeren- tial equation. For instance, it can be quite useful in estimating the eigenvalues. In this study we present a new technique for investigation some computational aspects of the eigenvalues. Particularly, we give an operator-pencil formulation of the problem and establish lower bound estimation for eigenvalues by using modiﬁed Rayleigh quotient.

Keywords: Boundary value problems, boundary and transmission conditions, eigenvalues, generalized eigenfunctions, Riesz basis.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 66

Source Identification Problems for SchrödingerDifferential and Difference Equations Allaberen Ashyralyev1,a,b,c, Mesut Urun2,a,b 1a Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. 1bFriendship’ University of Russia (RUDN University),Aˆ Ul Miklukho Maklaya 6, Moscow 117198, Russia. 1c Institute of Mathematics and Mathematical Modeling, 050010, Almaty, Kazakhstan. [email protected] 2a Istanbul Technical University, Istanbul, Turkey. 2b Murat Education Institution, Istanbul, Turkey. [email protected]

Abstract: In the present study, a source identification problem for a one-dimensional Schrodinger equation is investigated. Stability estimates for the solution of the source identification problem are established. A first order of accuracy difference scheme for the numerical solution of this problem is presented. Stability estimates for the solution of this difference scheme are established. Numerical results are provided. Keywords: Source identification problem, Schrodinger differential equation, difference scheme. 2010 Mathematics Subject Classification: 35Q41, 65M06, 34G10 References: [1] S. I. Kabanikhin, Methods for solving dynamic inverse problems for hyperbolic equations, J. In- verse Ill-Posed Probl.,12: 493-517, 2014. [2] Yu.Ya. Belov, Inverse problems for partial differential equations, Inverse and Ill-posed Problems Series -VSP, 2002. [3] A.Ashyralyev, C. Ashyralyyev, On the problem of determining the parameter of an elliptic equation in a Banach space, Nonlinear Analysis: Modelling and Control, 19(3): 350-366, 2014. [4] A. Ashyralyev, M. Urun, Determination of a control parameter for the difference Schrodinger equation, Abstract and Applied Analysis, Article Number: 548201, DOI: 10.1155/2013/548201, 2013. [5] I. Orazov and M. A. Sadybekov, Siberian Math. J. 53:146-151, 2012.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 67

Stability of the third order partial diﬀerential equations with time delay Allaberen Ashyralyev1, Evren Hincal2, Suleiman Ibrahim3 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey Peoples’ Friendship University of Russia (RUDN University), Ul Miklukho Maklaya 6, Moscow 117198, Russia Institute of Mathematics and Mathematical Modeling, 050010, Almaty,Kazakhstan [email protected] 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey [email protected] 3 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey [email protected]

Abstract: In the present paper, the initial value problem for the third order partial diﬀerential equations with time delay in a Hilbert space with self-adjoint positive deﬁnite operator is investigated. The main theorem on stability of this problem is established. The application of this theorem is presented. Additionally, illustrative numerical results are provided.

Keywords: Partial diﬀerential equation, stability, time delay, diﬀerence scheme

2010 Mathematics Subject Classification: 35G10, 65M12, 34D20 References: [1] Y. Apakov, S. Rutkauskas, On a boundary value problem to third order pde with multiple characteristics, Nonlinear Analysis-Modelling and Control, vol. 16, no 3, 255–269, 2011. [2] A.L. Skubachevskii, On the problem of attainment of equilibrium for control-system with delay, Doklady Akademii Nauk, vol. 335, no 2, 157–160, 1994. [3] A. Ashyralyev, S.N. Simsek, An operator method for a third order partial differential equation, Numerical Functional Analysis and Optimization, vol. 38, no 10, 1341–1359, 2017. [4] A. Ashyralyev, Kh. Belakroum, and A. Guezane-Lakoud, Stability of boundary value problems for third-order partial differential equations, Electronic Journal of Differential Equations, vol. 2017, no 53, 1–11, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 68

Numerical solutions of the system of partial diﬀerential equations for observing epidemic models A. Ashyralyev1, E. Hincal2, B.Kaymakamzade3 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey Peoples’ Friendship University of Russia (RUDN University), Ul Miklukho Maklaya 6, Moscow 117198, Russia Institute of Mathematics and Mathematical Modeling, 050010, Almaty,Kazakhstan [email protected] 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey [email protected] 3 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey [email protected]

Abstract: In the present paper, stability of initial-boundary value problem for the system of partial diﬀerential equations for observing HIV mother to child transmission epidemic models is studied. Applying operator approach, theorems on stability of this problem and of diﬀerence schemes for approximate solutions of this problem are established. The generality of the approach considered in this paper, however, allows for treating a wider class of multidimensional problems. Numerical results are provided.

Keywords: Diﬀerence schemes, system of partial diﬀerential equation, epidemic models, realization in computer.

2010 Mathematics Subject Classification: 35K40, 65M12, 92B05 References: [1] Alaa Elkadry, Transmission Rate in Partial Differential Equation in Epidemic Models, Master Thesis, Marshall University, 2013. [2] A.A. Samarskii, The Theory of Difference Schemes, CRC Press; 1st edition, 2001. [3] A. Ashyralyev, P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations,Operator Theory Advances and Applications, BirkhäuserVerlag, Basel, Boston, Berlin, 2004. [4] S. G. Krein, Linear Differential Equations in Banach space, Nauka: Moscow,1966.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 69

A Mathematical Cancer Model with BCG Immunotherapy Combined with Immune Checkpoints Inhibitors: an Optimal Control Approach

Evren Hincal1, Farouk Tijjani Saad2, Isa Abdullahi Baba3 1 Department of Mathematics, Near East University, Mersin 10, TRNC. [email protected] 2 Department of Mathematics, Near East University, Mersin 10, TRNC. [email protected] 3 Department of Mathematics, Near East University, Mersin 10, TRNC. [email protected]

Abstract: We present a mathematical model of cancer growth in the bladder that includes the immune cells, BCG, immune checkpoints and drug therapy (checkpoint inhibitor) in the form of a control function. The control function blocks the action of immune checkpoints on the immune system. Our aim here is to apply optimal control theory to ﬁnd a control strategy that will minimize the number of cancer cells in the bladder and cost of control. Existence of the optimal control is stated and Pontryagin’s maximum principle is used to characterize the nature of the control function. The optimality system obtained gives a two-point boundary value problem; hence, we use the forward-backward sweep method to present the numerical solutions of the system. The optimality conditions and characterization of the control are discussed.

Keywords: Optimal control, mathematical model, Hamiltonian, cancer, immune checkpoint, Pontryagin’s principle, forward-backward sweep method.

2010 Mathematics Subject Classification: 35J05, 35J08, 35J25 References: [1] Saad F. T., E. Hincal, and B. Kaymakamzade, Dynamics of immune checkpoints, immune system, and BCG in the treatment of Superficial bladder cancer, Computational and Mathematical Methods in Medicine, Volume 2017 (2017), Article ID 3573082, 9 pages. [2] W. H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975 [3] Thalya et al, Optimal control applied to immunotherapy, Discrete and continuous dynamical systems B, Volume 4, Number 1, February 2004. Pp 135-146. [4] A. Ghaffari, N. Naserifar, Optimal therapeutic protocols in cancer immunotherapy, Computers in Biology and Medicine 40 (2010) 261-270.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 70

An Interval Based Social Accounting Matrix for a Regional Economy

Behiye Cavusoglu1,Pinar Sharghi2,Serife Z. Eyupoglu3,Tulen Saner4 1 Near East University,Department of Economics, P.O.Box:99138, Nicosia, North Cyprus, Mersin 10 Turkey. [email protected] 2 Near East University,Department of Maritime Management, University of Kyrenia, Kemal Pars Street no: 1, Kyrenia, North Cyprus, Mersin 10 Turkey. [email protected] 3 Near East University, Department of Business Administration, P.O.Box:99138, Nicosia, North Cyprus, Mersin 10 Turkey. [email protected] 4 Near East University, School of Tourism and Hotel Management, P.O.Box:99138, Nicosia, North Cyprus, Mersin 10 Turkey. [email protected] Abstract The assessment of the major impact of economic events is based primarily on the input-output model. In all of the existing input-output models exact information is used. Yet, in real life, we can never usually ﬁnd such exact information for problems in particular related to the future. An expansion of the input-output model is the Social Accounting Matrix (SAM) and the SAM builds on Leontief’s input-output tables. In this paper we construct a SAM model for a regional economy and initial information is described by interval numbers. For the given region Northern Cyprus, where a scarcity of data exists, interval information based SAM, based on the best available information is suggested. Keywords: Input-output model, Social Accounting Matrix, interval number, regional economy References: [1] Aliev, R. A., Aliev, R. R. 2001.Soft Computing and Its Application. World Scientiﬁc. New Jersey, London, Singapore, Hong Kong. [2] Aliev, R. A.,Alizadeh, A. V.,Huseynov, O.H. 2015. The Arithmetic of Discrete Z-numbers. Inform. Sciences. Vol. 290, pp.134-155,http://dx.doi.org/10.1016/j.ins.2014.08.024.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 71

An Input-Output Analysis for a Regional Economy: The Case of North Cyprus

Behiye Cavusoglu1,Serife Z. Eyupoglu2,Tulen Saner3,Pinar Sharghi4 1 Near East University, Department of Economics, P.O.Box:99138, Nicosia, North Cyprus, Mersin 10 Turkey. [email protected] 2 Near East University, Department of Business Administration, P.O.Box:99138, Nicosia, North Cyprus, Mersin 10 Turkey. [email protected] 3 Near East University, School of Tourism and Hotel Management, P.O.Box:99138, Nicosia, North Cyprus, Mersin 10 Turkey. [email protected] 4 Near East University, Department of Maritime Management, University of Kyrenia, Kemal Pars Street no: 1, Kyrenia, North Cyprus, Mersin 10 Turkey. [email protected] Abstract Input-output analysis is an economic tool that is used in order to measure the impact of an existing, planned or expected business operation, decision or event on the economy. This paper presents an Input-output Analysis with the latest available information for a national economy, which is an important source of information to understand the inter-relations existing among the different sectors of an economy. This paper focuses on the North Cyprus economy and the input-output analysis was used to measure the impact of changes in two sectors, namely Tourism and Education on the national economy. Keywords: Input-output analysis, input-output table, Education, Tourism, North Cyprus References: [1] Douglas, A. J., David A. H. (1995). Estimating Recreation Employment Effects with IMPLAN for the Glen Canyon Dam Region. Journal of Environmental Management, 44, 233-247. [2] Krikelas, A. C. (1992). Why Regions Grow: A Review of Research on the Economic Base Model. Federal Reserve Bank of Atlanta Economic Review 77, 16-29. [3] Miller, R.E., and Blair, P.D., (1985). Input-Output Analysis: Foundations and Extensions. Prentice-Hall, Inc., Englewood Cliffs, New Jersey

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 72

Source Identification Problems for Two Dimensional Neutron Transport Differential and Difference Equations

Allaberen Ashyralyev 1,a,b,c, Abdulgafur Taskin 2 1aDepartment of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. 1bPeoples’ Frienship University of Russia (RUDN University), Ul Mikluko Maklaya 6, Moscow 117198, Russia 1cInstitute of Mathematics and Mathematical Modeling, 050010, Almaty, Kazakhstan. [email protected] 2Private Nakkastepe Bahcesehir Science and Technology College, Uskudar, Istanbul, Turkey. [email protected]

Abstract: In this study, an inverse problem concerning two dimensional neutron transport equation with a time-dependent source control parameter is considered. For the approximate solution of this problem a first order of accuracy difference scheme is constructed. Finite difference schemes are presented for identifying the control parameter. Stability inequalities for the solution of this problem are established. The results of a numerical experiment are presented, and the accuracy for this inverse problem is discussed.

Keywords: Source identiﬁcation problem, Two dimensional neutron transport equation, diﬀerence scheme.

2010 Mathematics Subject Classification: 65M12, 65M06, 82D75 References: [1] A. Ashyralyev, and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser,Verlag, Berlin, Germany, 2004. [2] A. Ashyralyev, and A. Taskin, AIP Conference Proceedings 1389, 570 (2011), (DOI: 10.1063/1.3636794). [3] A. Ashyralyev, and A. Taskin, AIP Conference Proceedings 1676, 020051 (2015), (DOI: 10.1063/1.4930477). [4] A. S. Erdogan, A. Sazaklioglu, A note on the numerical solution of an identification problem for observing two-phase flow in capillaries, Mathematical method in the Aplied Sciences, 37 (16) (2014), pp. 23932405. [5] A. A. Samarskii and P.N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Math- ematical Physics, Inverse and Ill-posed Problems Series,Walter de Gruyter, Berlin, Germany, 2007.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 73

New type Sturm-Liouville eigenvalue problem with interior singular point

Kadriye Aydemir1, Hayati Olˇgar 2 and Oktay Sh. Mukhtarov 2,3 1 Department of Mathematics, Faculty of Science Art, Amasya University, Turkey E-mail: [email protected] 2 Department of Mathematics, Faculty of Science Art, Gaziosmanpa¸saUniversity, Turkey E-mail: [email protected] 2,3Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Azerbaijan E-mail: [email protected]

Abstract: This work investigates a kind of discontinuous Sturm-Liouville problem with boundary conditions, and transmission conditions are imposed at an singular interior point. We establish some spectral properties of the considered problem. At ﬁrst, we shall deﬁne some new Hilbert spaces and give some inequalities which is needed for further investigation. Then, we suggest some methods for solving these Sturm-Liouville problems with transmission conditions.

Keywords:Sturm-Liouville problems, eigenvalues, transmission conditions.

2010 Mathematics Subject Classification: 34L10, 34L15 Acknowledgement:This work was supported by Amasya University Research Fun for financial support through Project number FMB-BAP 18-0323 (The Scientific Re- search Projects Coordination Unit).

References: [1] I. M. Krichever, Spectral theory of two-dimensional periodic operators and its applications. Rus- sian mathematical surveys, 44: 145, 1989. [2] M. S .P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic Press), 1973. [3] D. D. Bainov, P.S. Simeonov, Impulsive differential equations: periodic solutions and applications, Hacloi: Longman Scientific and Technical, 1993. [4] W. Ding, M. Han, Periodic boundary value problem for the second order impulsive functional differential equations, Applied Mathematics and Computation, 155, 709-726, 2004. [5] O. Sh. Mukhtarov, H. Olar, K. Aydemir, Resolvent Operator and Spectrum of New Type Boundary Value Problems. Filomat, 29(7), 1671-1680, 2015 .

New least-squares adjustments of nuclear binding energies of atomic nuclei

Djelloul Benzaid1, Salaheddine Bentridi1 1 Djilali Bounaama Khemis Milana University, Algeria [email protected] Abstract A new least-squares adjustment of binding energies of atomic nuclei based on the last updated atomic mass evaluation AME2016 has been considered. The new set of parameters of mass formula, obtained in this work, may reduce the relative error in binding energies when calculated in the framework of mass formula. A special treatment has been made for medium and heavy nuclei with atomic masses greater than 50. Keywords: Mass formula, least-squares adjustments, AME2016, binding energy

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 74

Adaptive conditional integrators sliding mode control scheme for a class of nonlinear systems

Abdelkader Kerraci1,3, Mohamed Faycal Khelﬁ 2, Ahmed-Foitih Zoubir3 1 Faculty of Science and Technology, Lesi Laboratory, University of Khemis Miliana, Algeria [email protected] 2 Faculty of Applied and Exact Sciences, RIIR Laboratory, University Ahmed Ben Bella Oran 1, Algeria mf khelﬁ@yahoo.fr 3 Faculty of Electrical Engineering, University of Sciences and Technology of Oran, Algeria [email protected]

Abstract: The present paper is dedicated to the presentation and implementation of conditional integrators sliding mode controllers based on adaptive approach in order to synthesise tracking errors and to ensure Lyapunov stability, of all system nonlinearities and desired tracking trajectories. The adaptive approach allows an on-line estimation of nonlinear system parameters to use them in conditional integrator sliding mode control scheme. Indeed the proposed control law needs the exact system model to give good performances. The complexity of the nonlinear systems makes it hard to know its parameters. However, the integrator of sliding mode controllers is introduced in such a way that it provides integral action only (conditionally), eﬀectively eliminating the performance degradation. Finally, the proposed controller design is illustrated by application to control single inverted pendulum with simulation in MATLAB environment.

Keywords: Nonlinear system, Sliding mode, Adaptive approche, Stability, Condition- nal integrators, Invert pundulum.

2010 Mathematics Subject Classiﬁcation: 34H15, 93C10, 93C40 References: [1] S. Seshagiri, H. K. Khalil, Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators, Automatica 41, 43–54, 2005. [2] F.Z. Daikh, M.F. Khelﬁ, Sliding mode with neuro-fuzzy network controller for inverted pendulum, Int. J. Automation and Control, Vol 9, no 1, 2015. [3] A. Kacimi, A. Mokhtari, B. Kouadri, Sliding mode control based on adaptive backstepping approch for quadrotor unmanned aerial vehicle, Electrical Review, ISSN 0033-297,R. 88 NR 6/2012

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 75

Review of modeling and simulation of natural nuclear reactors

Salaheddine Bentridi1, Benoit Gall2, Djelloul Benzaid1, Franois Gauthier-Lafaye3 1 Lab. of Energy and Smart Systems, University of Khemis-Miliana, Algeria 2 Institut Pluridisciplinaire Hubert Curien, UMR 7871, Strasbourg, France. 3 Laboratoire dHydrologie et de Gochimie de Strasbourg, France. [email protected]

Abstract: The present work intend to share the up-to-date modeling and simulation performed to explain the criticality occurrence in natural nuclear systems. Such systems were ignited 2 billion years ago, in the Oklo Uranium deposit, in Gabon (West Africa). The studied system is modelled by taking into account the geological context. The simulations were carried out with a Monte-carlo based code, known as MCNP; widely used for neutron physics and nuclear reactors calculation. By looking for the criticality conditions we resolved the following equation :

(2) keff (VUO2, φTOT , e, R, tB10) = 1 with: VUO2: volume fraction of uraninite, φTOT : totally saturated ore porosity, e, R: geometrical dimensions, tB10: Boron-10 equivalent content. Numerical simulations carried out with MCNP, generated a set of critical solutions as a parametrical curves, called Isocritical lines. Each point from such a line represent a possible natural critical configuration, given in correpsonding conditions of Oklo case. An isocritical line represent an interface in space parameters, separating a barren space (subcritical system) from fissile space (critical and supercritical systems). Although, any evolving natural nuclear reactor should at least reach the criticality line to start a dynamic which lead it to a new geological and physical state: Fuel burn-up, mineral alteration of the container ore, transmutation, fission products accumulation, etc... Detailed techniques used to model and simulate such systems are given in previous works [1].

Keywords: Oklo, Modeling and Simulation, MCNP, Monte-Carlo, Uranium, Natural Reactors References: [1] S.E. Bentridi & al. , Criticality of Oklo Natural Reactors: Realistic Model of the Reaction Zone 9 (Criticit des racteurs naturels dOklo : Modlisation raliste de la zone de raction RZ9) IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 60, NO. 1, FEBRUARY 2013

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 76

Delay Epidemic Model with and without Vaccine B. Kaymakamzade1, E. Hincal2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey [email protected] 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey [email protected]

Abstract: In this work, two models with and without vaccine are constucted. For both model basic reproduction numbers are founded as a e−µτ βΛ e−µτ Λ kr R1 = ,R2 = β + . 0 γ + µ + d 0 (γ + µ + d)(µ + r) µ + 1 − k

From the basic reproduction ratios it can be seen that when there is no vacccine, disease can only controled with reducing the infectious rate β or decreasing the incubation period. However, when there is a vaccine it is enough to increase the rate of the vaccine r to controlled the disease. In addition to this if transmission rate of infectious also decrease then the disease will disappear faster. Delay effect on this model is very rare. When we are talking about delay on this paper we mean that incubation period. If we have enough vaccine the effect of delay is very tiny. However if there is no vaccine you can see the effect of delay. Two equilibria which are disease free and endemic equilibriums are found and with using Lyapunov function it is shown that the global stabilities of each equilibria for both model. For the first model it is found that DFE E0 is globally asymtotically stable when 1 R0 < 1 and endemic equilibrium E1 is always asymtotically stable. With using similar 2 method it is shown that E0 is asymtotically stable when R0 < 1 and E1 is always global asymtotically stable for model 2. In last section numerical simulations are given for both model.

Keywords: Delay, Epidemic, Modelling, Vaccine diﬀerential equation.

2010 Mathematics Subject Classiﬁcation: 35N05, 37C75, 34K50 References: [1] S.Chauhan, O.P. Misra, J.Dhar. American Journal of Computational and Applied Mathematics 2014, 17-23 [2] B.Kaymakamzade, E.Hincal. Quality and Quantity DOI 10.1007/s11135-017-0647-812 2017 [3] M.E. Alexander, C. Bowman, S.M. Moghadas, R. Summers, A.B. Gumel, B.M. SahaiA vaccination model for transmission dynamics of inﬂuenza SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 503-524 [4] B. Buonomo, A. dOnofrio, D. LacitignolaGlobal stability of an SIR epidemic model with information dependent vaccination Math. Biosci., 216 (2008), pp. 9-16

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 77

Principle of Independence of Continuation of Functions k- Signiﬁcant Logic From Coding Anvar Kabulov1,Ibrokhimali Normatov2,Ilyos Kalandarov3 1 National University of Uzbekistan [email protected] 2 Tashkent University of Information Technologies named after Muhammed Al-Khwarızmi, Uzbekistan ibragim [email protected] 3 Navoi State Mining Institute, Uzbekistan [email protected] Abstract: In this paper we consider not everywhere deﬁned functions of k-valued logic. The problem of logical continuation of functions of k-valued logic in the class of disjunctive normal forms is investigated. We prove a theorem on the invariant extension of functions of k-valued logic in the class of disjunctive norms, which does not depend on the accepted coding. An algorithm is constructed for constructing a set of invariant points that are independent of the received encoding. Keywords: Encoding, logic, conjunction, disjunction, sets, quasi-Boolean invariant, synthesis, equivalent n We consider the function F (x1, x2, ... , xn) k - valued logic, given on M ⊆ Ek : F (˜x) = γj , ifx ˜ ∈ Mj , j = 0, k − 1 ,

m S where M = Mi and Mi ∩ Mj = ∅ for i 6= j. We denote by {π} the set of all i=0 permutations of the set {0, 1, ..., k − 1} : π = (i0, i1, ..., ik−1) . The functions Fπ (˜x) = ij , ifx ˜ ∈ Mj , j = 0, k − 1 , we call π is a permutation of F (˜x). We will say that n a pointα ˜ ∈ Ek \M preserves the encoding (code) of a set Mj ⊆ M, j = 0, k − 1 with respect to a permutation pi , if MΣTF (˜α) = j and MΣTFπ (˜α) = ij . A point n α˜ ∈ Ek \M is called a point that preserves the encoding of the set Mj , if it preserves the encoding Mj with respect to any permutation π ∈ {π} . Let M = V MΣTFπ . π:π∈(π)

k−1 n S Theorem. The pointα ˜ ∈ Ek \ Mi preserves the code of the set Mj , then and i=0 only if, when:

1) in DNF M there is an EK A such that NA ∩ Mj 6= ∅, α˜ ∈ NA,NA ∩ Mi = ∅, where (i = 0, ..., j − 1, j + 1, ..., k − 1) ;

2) each interval NA where EK A belongs to the set M intersects with Mi, (i 6= j) and contains the point Mj . References: [1] Yu.I. Zhuravlev, Algorithms for constructing minimal disjunctive normal forms for a function of the algebra of logic, Discrete mathematics and mathematical problems of cybernetics, vol.1, Moscow: Nauka, 1974, p.67-98..

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 78

Diﬀerence schemes for delay Schr¨odingerequation

Deniz Agirseven1 1 Department of Mathematics, Trakya University, Edirne, Turkey [email protected]

Abstract: In the present paper, the first and second order of accuracy stable difference schemes for the Schrödinger equation with time delay in a Hilbert space are presented [1]. Theorems on stability estimates for the solution of these difference schemes are established by using approaches of [2]. The application of theorems on stability of the approximate solution of difference schemes for the three types of delay Schrödinger partial differential problems is provided. Additionally, some illustrative numerical results are given.

Keywords: Schrödinger differential equations, difference schemes, stability estimates

2010 Mathematics Subject Classification: 65M06, 35R11 References: [1] D. Agirseven, On the stability of the Schrödingerequation with time delay, Filomat 32:3, 759-766, 2018. https://doi.org/10.2298/FIL1803759A [2] A. Ashyralyev, P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Op- erator Theory Advances and Applications, Birkhauser Verlag, Basel, Boston, Berlin, 2004.

On numerical solution of the Schr¨odinger-parabolic equations

Yildirim Ozdemir1, Aylin Aygul Turkyilmaz1 1 Department of Mathematics, Duzce University, Turkey [email protected]

Abstract: The nonlocal boundary value problem for Schrödinger-parabolic equation is considered. The stability estimates for the solution of the given problem is established. The first and second order of difference schemes are presented for approximately solving a specific nonlocal boundary problem. The theoretical statements for the solution of these difference schemes are supported by the result of numerical examples.

Keywords: Nonlocal boundary value problem, Diﬀerence scheme, Stability.

2010 Mathematics Subject Classification: 34B10, 65L12, 35B35 References: [1] Y. Ozdemir and M. Alp, AIP Conf.. Proc. 1611, 221-224 (2014). [2] B. Hicdurmaz and A. Ashyralyev, Numer. Func. Anal. Opt. 38, 1215-1225 (2017). [3] A. Ashyralyev and Y. Ozdemir, Taiwan. J. Math. 4, 1075-1089 (2007). [4] A. Ashyralyev, AIP Conf. Proc. 1880, (2017). [5] A. Ashyralyev and O. Gercek, Discrete. Dyn. Nat. Soc. 2008, 1-16 (2008). [6] A. Ashyralyev and B. Hicdurmaz, Kybernetes 40, 736-750 (2011). [7] A. Ashyralyev and Y. Ozdemir, Comput. Math. Appl. 50, 1443-1476 (2005). [8] P. E. Sobolevskii, Difference Methods for Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, 1975, (in Russian).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 79

On the numerical solution of nonlinear system of coupled sine-Gordon equations

Ozgur Yildirim1, Meltem Uzun2, Oguz Altun3 1 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey E-mail: [email protected] 2 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey E-mail: [email protected] 3 Department of Computer Engineering, Yildiz Technical University, Istanbul, Turkey E-mail: [email protected]

Abstract: In this paper, a system of nonlinear coupled sine-Gordon, which have some powerful applications in physics and biology is considered. A special case of this system, which describe the open states in DNA double helices is studied. Numerical solution of this system is obtained by finite difference method with fixed point iteration. Some examples are considered and the results of numerical experiments are presented.

Keywords: nonlinear boundary value problems, diﬀerence equations, numerical analysis

2010 Mathematics Subject Classification: 34B15, 39A10, 97N40 References: [1] Khusnutdinova K.R., Pelinovsky D.E., ”On the exchange of energy in coupled Klein–Gordon equations”, Wave Motion 38 (2003) 1–10 (2003). [2] Ashyralyev A., Yildirim O., ”On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations”, Taiwanese Journal of Mathematics 14(1),165–194 (2010). [3] Ashyralyev A., Agirseven D., and Ceylan B., ”Bounded solutions of delay nonlinear evolutionary equations”, Journal of Computational and Applied Mathematics 318, 69–78 (2017).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 80

A note on stable diﬀerence schemes for a third order partial diﬀerential equation

Allaberen Ashyralyev1, Kheireddine Belakroum2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey, Friendship’ University of Russia (RUDN University),Aˆ Ul Miklukho Maklaya 6, Moscow 117198, Russia, and Institute of Mathematics and Mathematical Modeling, 050010, Almaty, Kazakhstan [email protected] 2Department of Mathematics, Freres Mentouri University, Constantine, Algeria [email protected]

Abstract: Boundary value problems for third order partial differential equations have been a major research area in thermal conductivity [1], micro-scale heat transfer [2] and modern physics. The well-posedness of various nonlocal and local boundary value problems for partial differential and difference equations have been studied ex- tensively by many researchers (see [1, 1–4, 6–8] and the references given therein). In this study, the nonlocal boundary value problem for the third order partial differential equation with a self-adjoint positive definite operator in a Hilbert space is studied. The stability estimates for the solution of the problem are established. In applications, the stability estimates for the solution of two types of third order partial differential equations are obtained. A stable three-step difference schemes for the approximate solution of the problem is presented. The theorem on stability of these difference schemes is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of nonlocal boundary value problem for third order partial differential equations are obtained. For the numerical analysis, a first and a second order of approximation difference schemes for a one dimensional third order partial differential equation are presented. Moreover, some numerical results are given.

Keywords: Nonlocal boundary value problems, Stability, Difference scheme,Third order partial differential equation, Self-adjoint positive definite operator, Hilbert space.

2010 Mathematics Subject Classification: 35G15; 47A62 References: [1] A. Ashyralyev. A note on the bitsadze-samarskii type nonlocal boundary value problem in a banach space. Journal of Mathematical Analysis and Applications, 344(1):557–573, 2008. DOI:10.1016/j.jmaa.2008.03.008. [2] A. Ashyralyev, I. Karatay, and P. E. Sobolevskii. On well-posedness of the nonlocal boundary value problem for parabolic difference equations. Discrete Dynamics in Nature and Society, 2:273–286, 2004. DOI: 10.1155/S1026022604403033. [3] A. Ashyralyev, Kh. Belakroum, and A. Guezane-Lakoud. Stability of boundary-value problems for third-order partial differential equations. Electronic Journal of Differential Equations, 2017(53):1– 11, 2017. [4] Kh. Belakroum, A. Ashyralyev, and A. Guezane-Lakoud. A note on the nonlocal boundary value problem for a third order partial differential equation. In AIP Conference Proceedings, volume 1759, page 020021. AIP Publishing, 2016. [5] A. U. Coskun. Investigation of microscale heat transfer e¨ıfects in nanoscale electronic devices by spectral methods. Abstract Book of International Conference on Mathematical Modeling and Scientific Computing, 2001. [6] N. Gordeziani, P. Natani, and P. E. Ricci. Finite difference methods for solution of nonlocal boundary value problems. Computers and Mathematics with Applications, 50:1333–1344, 2005. [7] F. Ivanauskas, T. Meskauskas, and M. Sapagovas. Stability of difference schemes for two- dimensional parabolic equations with non-local boundary conditions. Applied Mathematics and Computation, 215(7):2716–2732, 2009. [8] F. F. Ivanauskas, Yu. A. Novitski, and M. P. Sapagovas. On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions. Differential Equations, 49(7):849–856, 2013. [9] V. I. Korzyuk and V. B. Kiselev. A boundary value problem for a third-order linear nonclassical equation generated by the heat equation. Vest. Nats. Akad. Navuk Belaruser. Fiz.-Mat. Navuk, 140(3):5–11, 2001. in Russian.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 81

Invariant diﬀerence schemes for sine-Gordon equations

Ozgur Yildirim1, Sumeyra Caglak2 1 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey E-mail: [email protected] 2 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey E-mail: [email protected]

Abstract: In this paper, we consider invariant differential and present corresponding invariant difference equations which preserve the Lie group symmetries of the sine- Gordon equation. We present invariant difference schmes for the approximate solution of sine-Gordon equation which conserve uniformity and orthogonality of meshes.

have some powerful applications in physics and biology. A special case of this system, which describe the open states in DNA double helices is studied. Numerical solution of this system is obtained by finite difference method with fixed point iteration. Some examples are considered and the results of numerical experiments are presented.

Keywords: nonlinear boundary value problems, diﬀerence equations, numerical analysis

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 82

Uniﬁcation of q−exponential function and related q−numbers and polynomials Nazim I. Mahmudov 1, and Mohammad Momenzadeh 2 1 Eastern Mediterranean University Gazimagusa, TRNC, Mersin 10, Turkey

2 Near East University Nicosia, TRNC, Mersin 10, Turkey

E-mail: 1 [email protected], 2 [email protected]

Abstract: The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the generating function. we unify all forms of q−exponential functions by one more parameter. we study some conditions on this parameter to related this to some classical results for q-Bernoulli numbers and polynomials. This uniﬁcation reach to the nonlinear system of equations for that extra parameter and give us the general form of symmetric q- exponential function.

Keywords: q-Exponential function, q-calculus, q-Bernoulli polynomials, q-Euler number. References: [1] Exton, H, q-Hypergeometric Functions and Applications, Halstead Press, New York, USA, 1983. [2] Jan L. Cie´sli´nski, Improved q-exponential and q-trigonometric functions,Applied Mathematics Letters Volume 24, Issue 12, December 2011, Pages 2110–2114. [3] H. M. Srivastava, T. Kim and Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-Zeta functions and basic L-series, Russian J. Math. Phys., 12 (2005), 241-268. [4] Q.-M. Luo, H.M. Srivastava, q-extensions of some relationships between the Bernoulli and Euler polynomials, Taiwanese Journal Math., 15, No. 1, pp. 241-257, 2011. [5] M. Cenkci and M. Can, Some results on q-analogue of the Lerch Zeta function, Adv. Stud. Con- temp. Math., 12 (2006), 213-223. [6] J. Choi, P. J. Anderson and H. M. Srivastava, Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of q-Hurwitz zeta functions, Appl. Math. Comput., 215 (2009), 1185- 1208. [7] J. Choi, P. J. Anderson and H. M. Srivastava, Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz Zeta function, Appl. Math. Comput., 199 (2008), 723-737. [8] N. I. Mahmudov and M. Momenzadeh, On a Class of q-Bernoulli, q-Euler, and q-Genocchi Poly- nomials, Hindawi, Abstract and Applied Analysis, Volume 2014 (2014).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 83

Suitable variations for degenerate modes of the simplex method

Yuriy Seisov1, Yuldashbay Kurambayev2 1International University of Oil and Gas, Turkmenistan 2Oguz han Engineering and Technology University of Turkmenistan, Turkmenistan 1 [email protected], 2k [email protected] The problem of ﬁnding a nontrivial solution of problem  a x + a x + ... + a x ≤ 0,  11 1 12 2 1n n (1) ...  am1x1 + am2x2 + ... + amnxn ≤ 0 at observance of restrictions

(2) xi ≥ 0, i = 1, n.

The question of existence of the untrivial solution of this task is key in case of the degenerate modes at the solution of tasks of the linear programming (LP) by simplex method [1]. It is shown in work [2]. Here we will specify an eﬀective method of ﬁnding of the solution of an objective if it exists. T Let Aj = (a1j , . . . , amj ) be normalized vector. Let us v u m uX 2 (3) kAk = t aij = 1 i=1

It does not reduce a community of consideration of the considered task [2]. Having entered in addition m of variables xi ≥ 0, i = n + 1, n + m we will transform inequalities (1) to system of the linear equations

 a x + a x + ... + a x + x = 0,  11 1 12 2 1n n n+1 (4) ...  am1x1 + am2x2 + ... + amnxn + xn+m = 0.

The general restriction for all variables now have an appearance

(5) xi ≥ 0, i = 1, n + m

Denote by An+k for any k = 1, m vector column m × 1 with zero elements in each row expect kth row whode element is one. Then, we can write system (4) in a vector form n+m X (6) xj Aj = 0 j=1

We form a set of U in m - measured Euclidean space of Rm with elements u = T (u1, u2, . . . , um) such that

(7) uk = a11x1 + a12x2 + ... + a1nxn + xn+k, k = 1, m representing any linear combinations of vectors Aj , j = (1, n + m) , with the nonnegative coefficients of xj satisfy the condition n+m X (8) xi = 1 i=1 We will note that the system (4) is uniform; therefore the condition (8) doesn’t reduce a community of consideration of a task [2]. Geometrically set of U represents a polyhedron [1] in Rm space, with vertices of Aj , j = (1, n + m), located on the sphere S with single radiuses. It is known that any Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 84 point of u limited polyhedron in Rm is representable in the form of a linear combination but at most of m + 1 â“its vertices n+m X (9) u = xj Aj , j=1 at the same time restrictions (5) and (8) are observed. Thus, if (10) 0 ∈ U, Pn+m that representation takes place 0 = j=1 xj Aj , so the conditions (5), (6) and (8) are satisfied.

Theorem: In order that the initial task (1),(2) had the decision enough performance of inclusion (10). This theorem has allowed to make eﬀective algorithm and the program of the solution of LP tasks with the degenerate modes [2]. Made a comparison of the proposed method with known modiﬁcations of the simplex method [2].

2010 Mathematics Subject Classiﬁcation: 35J05, 35J08, 35J25 References: [1] Yudin D.B., Golshteyn E. G. Lineyni programmirovaniye (teoriya, metody i prilojeniya), - M.: Nauka, 1969 (in Russian). [2] Seisov Yu.B., Geldiyev H.A., Amanov A.T., Kurambayev Y. - Vyrojdenniye rejimi lineynogo programmirovaniya.- M.: Izdatelstvo ” Sputnik” , 2015, 104 s (in Russian).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 85

Numerical solutions of source identiﬁcation problem for hyperbolic-parabolic equations

Maral Ashyralyyeva1, Maksat Ashyraliyev2 1 Department of Applied Mathematics and Informatics, Magtymguly Turkmen State University, Turkmenistan 2 Department of Software Engineering, Bahcesehir University, Turkey [email protected]

Abstract: Numerical methods for partial differential equations with unknown source terms have been studied recently by many researchers (see for instance [1–3] and the references therein). In this work, a numerical study is presented of boundary value problem for hyperbolic- parabolic equations  u (t, x) − a(x)u (t, x) = p(x) + f(t, x), 0 < x < 1, 0 < t < 1,  tt x x    ut(t, x) − a(x)ux(t, x) = p(x) + g(t, x), 0 < x < 1, −1 < t < 0,  x  (1) u(0+, x) = u(0−, x), ut(0+, x) = ut(0−, x), 0 ≤ x ≤ 1,    u(−1, x) = ϕ(x), u(1, x) = ψ(x), 0 ≤ x ≤ 1,    u(t, 0) = u(t, 1) = 0, −1 ≤ t ≤ 1. where p(x) is an unknown source term. The first and second order of accuracy difference schemes for the approximate solution of problem (1) are constructed and the numerical procedure for implementation of these schemes is discussed. These difference schemes are implemented for a simple test problem and the numerical results are presented.

Keywords: Diﬀerence scheme, source identiﬁcation problem, hyperbolic-parabolic equation

2010 Mathematics Subject Classification: 65M06, 65N05, 35M13, 35R30 References: [1] A. Ashyralyev, A.S. Erdogan, O. Demirdag, On the determination of the right-hand side in a parabolic equation, Applied Numerical Mathematics, vol. 62, 1672–1683, 2012. [2] A.U. Sazaklioglu, A. Ashyralyev, A.S. Erdogan, Existence and uniqueness results for an inverse problem for semilinear parabolic equations, Filomat, vol. 31:4, 1057–1064, 2017. [3] M.A. Ashyralyyeva, M. Ashyraliyev, On a second order of accuracy stable difference scheme for the solution of a source identification problem for hyperbolic-parabolic equations, AIP Conference Proceedings, vol. 1759, 0200231–0200236, 2016.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 86

On the stable diﬀerence scheme for the time delay telegraph equation Allaberen Ashyralyev1, Deniz Agirseven2, Koray Turk2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Peoples’ Friendship University of Russia (RUDN University), Moscow , Russia. Institute of Mathematics and Mathematical Modeling 050010, Almaty, Kazakhstan. 2 Department of Mathematics, Trakya University, Edirne, Turkey. [email protected] [email protected] [email protected]

Abstract: In this study, the ﬁrst order of accuracay diﬀerence scheme

 uk+1−2uk+uk−1 uk+1−uk + α + Au = aAuh i ,  τ 2 τ k+1 k  N+1 N 1 ≤ k < ∞, Nτ = 1, (m − 1)N ≤ k ≤ mN, m = 1, 2, ...   2 u1−u0 u0 = ϕ, ((1 + ατ)I + τ A) τ = ψ is considered for the approximate solution of the initial value problem for the telegraph equation with time delay

( 2 d u(t) + α du(t) + Au(t) = aAu([t]), 0 < t < ∞, dt2 dt u(0) = ϕ, u0(0) = ψ in a Hilbert space H with a self-adjoint positive deﬁnite operator A, A > δI. Here ϕ and α2 ψ are elements of D(A) and [t] denotes the greatest-integer function and δ > 4 . The stability of this diﬀerence scheme is obtained and numerical results for a test problem are presented.

Keywords: Telegraph equation; time delay; diﬀerence schemes; stability.

2010 Mathematics Subject Classification: 35L10, 65N06. References: [1] A. Ashyralyev, P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, BirkhäuserVerlag, Basel, Boston, Berlin, 2004 [2] A. Ashyralyev, D. Agirseven, K. Turk, On the stability of the telegraph equation with time delay. AIP Conference Proceedings 1759, 020022 (2016); doi: 10.1063/1.4959636.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 87

Numerical solutions of source identiﬁcation problem for hyperbolic-parabolic equations

Keywords: Diﬀerence scheme, source identiﬁcation problem, hyperbolic-parabolic equation

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 88

On the stable diﬀerence scheme for the nonlocal fractional PDE Allaberen Ashyralyev1, Zafer Cakir2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Peoples’ Friendship University of Russia (RUDN University), Moscow , Russia. Institute of Mathematics and Mathematical Modeling 050010, Almaty, Kazakhstan. 2 Department of Mathematical Education, Alanya Alaaddin Keykubat University, Antalya, Turkey. [email protected] [email protected]

Abstract: In the present study, we consider the nonlocal boundary value problem for fractional parabolic equations. Firstly, we construct the first order of accuracy difference scheme for solution of nonlocal boundary value problem using the self-adjoint positive defined operator A. Then, the stability estimates for the solution of these difference scheme and some theoretical results are given.

Keywords: Nonlocal BVP; fractional PDE; diﬀerence schemes; stability.

2010 Mathematics Subject Classification: 35R11, 34K37. References: [1] A. Ashyralyev, P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, BirkhäuserVerlag, Basel, Boston, Berlin, 2004. [2] A. Ashyralyev, A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space. Journal of Mathematical Analysis and Applications 344, 557–573 (2008); doi: 10.1016/j.jmaa.2008.03.008. [3] A. Ashyralyev, O. Yildirim, On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations. Taiwanese Journal of Mathematics 14, 165–194 (2010).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 89

New phenomenas for degenerate nonlinear cross system with convective transfer and absorption M. Aripov1, M. B. Khojimurodova2 1 National University of Uzbekistan, Uzbekistan, [email protected] 2 Tashkent University of Information Technologies, Uzbekistan, [email protected]

Abstract: The purpose of this article is to investigate properties of the process of a nonlinear diﬀusion-reaction with heterogeneous density in the domain Q = (t, x): t > 0, x ∈ RN : p−2 ∂u m1−1 k β1 ∂t = div v ∇u ∇u − div(c(t)u) − γ1(t)u , (1) p−2 ∂v m2−1 k β2 ∂t = div u ∇v ∇v − div(c(t)v) − γ2(t)v ,

N (2) u (0, x) = u0 (x) ≥ 0, v (0, x) = v0 (x) ≥ 0, x ∈ R where k ≥ 1, p, mi , βi, i = 1, 2 - given positive numbers, ∇(.) − grad(.), functions x N u0 (x) , v0 (x) ≥ 0, x ∈ R , 0 < γi(t) ∈ C(0, ∞), i = 1, 2. The system (1) describes a set of physical processes, for example process of mutual reaction - diffusions, heat conductivity, a polytropical filtration of a liquid and gas in β β the nonlinear environment whose capacity equal to γ1(t)u 1 , γ2(t)v 2 . Particular cases (k = 1, p = 2) of the system were considered in works (see for instance [1-3]). In this work is established the estimate of a weak solution, the Fujita type critical exponent, a global solvability of solution, the phenomena of a finite speed of perturbation of disturbance, space localization, localized a wave type structure. References: [1] Samarskii A.A., Galaktionov V.A., Kurduomov S.P., Mikhailov A.P. Blowe-up in quasilinear parabolic equations, Berlin, 4, Walter de Grueter, 535 (1995). [2] Aripov M., Sadullaeva Sh.A., An asymptotic analysis of a self-similar solution for the double nonlinear reaction-diffusion system,J.Nanosystems: physics, chemistry, mathematics, 6 (6), 793-802 (2015). [3] Aripov M., Sadullaeva Sh.A. Qualitative properties of solutions of a doubly nonlinear reaction- diffusion system with a source, Journal of Applied Mathematics and Physics,3, 1090-1099 (2015).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 90

Mathematics Education

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 91

Problems of heat conduction with diﬀerent boundary conditions in noncylindrical domains

Minzilya T. Kosmakova, Nurgul T. Orumbayeva, Nurbulat K. Medeubaev, ZHanar M. Tuleutaeva Buketov Karaganda state university, Kazakhstan Svetik [email protected], [email protected] [email protected], [email protected]

Abstract: In a degenerating domain G = {(x; t): t > 0, 0 < x < t} homogeneous problems for the heat equation with diﬀerent boundary conditions are considered. Problems with conditions one of which contains the time derivative are reduced to singular integral equations with incompressible kernel: t Z (1) ϕ(t) − K(t, τ) ϕ(τ) dτ = 0, 0 where ( ! ) 1 t + τ (t + τ)2 1 t − τ K(t, τ) = √ − exp − + exp − . 3 2 1 2 2a π (t − τ) 2 4a (t − τ) (t − τ) 2 4a

It is shown that the norm of the integral operator in equation (1), acting in classes of continuous functions, is equal to 1. The integral equation for other BVP is considered in [1]. Its solution was found in an explicit form. It is shown that the original homogeneous boundary value problems have nontrivial solutions in certain functional classes. The research and writing of this work was carried out a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant No. AP05132262).

Keywords: Volterra integral equation of the second kind, incompressibility of the kernel, degenerating domain

2010 Mathematics Subject Classiﬁcation: 35D05, 35K20, 45C05, 45D05 References: [1] Jenaliyev, Muvasharkhan; Amangaliyeva, Meiramkul; Kosmakova, Minzilya; Ramazanov Murat. On a Volterra equation of the second kind with ”incompressible” kernel, Advances in Diﬀerence Equations, 71, 14 p., 2015.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 92

Problems of heat conduction in noncylindrical domains with special boundary conditions containing the time derivative Muvasharkhan T. Jenaliyev1, Murat I. Ramazanov2, Sagyndyk A. Iskakov2 1 Institute of mathematics and mathematical modeling CSc MES RK, Kazakhstan [email protected] 2 Buketov Karaganda state university, Kazakhstan [email protected] [email protected]

Abstract: The paper researches pseudo - Volterra integral equations that arise in the study of the boundary value problem for the equation of heat conduction in a degenerating angular domain ∂u(x, t) ∂2u(x, t) (1) − a2 = g(x, t), {x, t} ∈ G = {x, t : −t < x < t, t > 0}; ∂t ∂x2

∂u(x, t) du˜1(t) ∂u(x, t) du˜2(t) (2) − + = h1(t), + = h2(t), ∂x dt ∂x dt x=−t x=t whereu ˜1(t) = u(−t, t), u˜2(t) = u(t, t). [1] The peculiarity of problem (1) – (2) lies in the fact that both boundaries are mobile and the boundary conditions contain derivatives with respect to time. It is shown that the corresponding homogeneous integral equations in the class of essentially bounded functions have nonzero solutions. The research and writing of this work was carried out a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant No. AP05132262).

Keywords: pseudo - Volterra integral equation, equation of heat conduction, angular domain

2010 Mathematics Subject Classiﬁcation: 35D05, 35K20, 45C05, 45D05 References: [1] Solonnikov V.A., Fasano A. One-dimensional parabolic problem arising in the study of some free boundary problems [in Russian] // Zapiski Nauchnykh Seminarov POMI. 2000. Vol.269. p.322–338.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 93

The uniqueness of the solution of the two-dimensional direct problem of a wave process with an instantaneous source and a ﬂat boundar

Satybaev A.Dj.1,a,, Anishenko Yu.V.1,b, Kokozova A.Zh.1,c, Alimkanov A.A.1,d 1 Department of Information Technology and Management, Osh Technological University, Kyrgyzstan [email protected], [email protected], [email protected], [email protected]

Abstract: This article is devoted to the justiﬁcation of the uniqueness of the solution of the problem. A two-dimensional direct generalized problem of hyperbolic type is reduced to a regular problem with data on characteristics and the uniqueness of the solution of this problem is shown. We consider the following problem: 0 00 b1(z, y) 0 σ1(z, y) σ1z(z, y) 0 utt(z, y, t) + ut(z, y, t) = ∆u(z, y, t) + uz(z, y, t) + c1(z, y) c1(z, y) c1(z, y) σ0 (z, y) 1y 0 a1(z, y) 2 (1) + uy(z, y, t) + u(z, y, t), (z, t) ∈ R+, y ∈ R, c1(z, y) c1(z, y)

0 1 (2) u(z, y, t)|t<0 ≡ 0, u (z, y, t)|z=0 = − (r(y)δ(t) + h(y)θ(t)), z 2 where a1(z, y), b1(z, y), c1(z, y), σ1(z, y) are the coeﬃcients of the equation describing the physical parameters of the medium, u(z, y, t) is the perturbation of the medium, 00 00 ∆u(z, y, t) = uzz + uyy is the Laplace’s operator, r(y), h(y) are source functions, δ(t) is the Dirac’s delta-function, θ(t) is the Heaviside’s theta-function.

Keywords: two-dimensional, direct problem, wave processes, source, instantaneous, ﬂat boundary, uniqueness of the solution

2010 Mathematics Subject Classification: 45K05, 35L10 References: [1] S.I. Kabanikhin, Incorrect and inverse problems, Novosibirsk, Siberian Scientific Publishing House, 2009. [2] V.G. Romanov, Stability of inverse problems, Moscow, The scientific world, 2004. [3] Yu.V. Anishenko, A.Dj. Satybaev, The Uniqueness of the Solution of the Two-dimensional Direct Problem of Geoelectrics with Instantaneous and Corded Sources, Science, New Technologies and Innovations 7 (2014) 25–29. [4] A.Zh. Kokozova, The Uniqueness of the Solution of the Two-dimensional Direct Problem of the Tele- graph Equation with Instantaneous and Corded Sources, News of Higher Educational Institutions of Kyrgyzstan 11 (2015) 19–23. [5] A.Dj. Satybaev, A.A. Alimkanov, The Uniqueness of the Solution of the Two-dimensional Direct Seismic Problem with Instantaneous and Corded Sources, Bulletin of the Kyrgyz State University of Construction, Transport and Architecture 4(54) (2016) 56–62. [6] A.Dj. Satybaev, G.A. Kaldybaeva, The Uniqueness of the Solution of the Direct Problem of Ther- moelasticity with a Flat Boundary, The Problems of Automation and Control 1 (2011) 59–63. [7] A.Dj. Satybaev, Zh.K. Matisakov, The Uniqueness of the Solution of a Direct Problem of Acoustics with an Instantaneous Source and a Flat Boundary, The Problems of Automation and Control (2010) 159–163.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 94

Numerical solution of a two-dimension direct problem of the wave process Satybaev A.Zh.1, Kokozova A.Zh.2, Anishenko Yu.V.3, Alimkanov A.A.4

1 Osh Technological University named after M.M.Adyshev,Osh, N.Isanova81str.,Kyrgyzstan [email protected] 2 Osh Technological University named after M.M.Adyshev,Osh, N.Isanova81str.,Kyrgyzstan [email protected]

Abstract: The problem of propagation of a potential along a nerve fiber, which is described by a telegraph equation of parabolic type, is considered.This problem is reduced to the inverse problem of the hyperbolic equation and the last problem is solved by a numerical finite-difference method, using methods:rectification of characteristics, isolation of features.A theorem on the convergence of an approximate solution to the exact solution of the inverse problem is proved.In connection with the equivalence of the problem, an approximate calculation of the membrane capacity of the problem is determined. Consider the following problem: 2 ∂ u(z,y,t) b1(z,y) ∂u(z,y,t) σ1(z,y) 2 + = 4u(z, y, t) ∂t c1(z,y) ∂t c1(z,y) σ (z,y) (1) + σ1z (z,y) ∂u(z,y,t) + 1y ∂u(z,y,t) + a1(z,y) u(z, y, t), c1(z,y) ∂z c1(z,y) ∂y c1(z,y) 2 (z, t) ∈ R+, y ∈ R,

∂u(z, y, t) 1 (2) u(z, y, t)|t<0 = 0, = − (r(y)δ(t) + h(y)θ(t)) ∂z z=0 2 Where a1, b1, c1, σ1 - coeﬃcients of the equation, meaning some physical parameters of the environment, u(z, y, t) - Environment pressure, r(y), h(y) - functions of environment sources, δ(t), θ(t) - Dirac delta function and theta function Heaviside, R+ = {x ∈ R : x ≥ 0}. The direct problem is to determine the pressure of the medium u(z, y, t) for known functions of the coeﬃcients of the equation and the function of the sources. In this paper, all these tasks are grouped, systematized, summarized results, and numerical solutions of the problem.

Keywords: Two-dimensional, direct problem, wave processes, instantaneous source, cord, numerical solution References: [1] S.I.Kabanichin, Incorrect and Inverse problem, Novosibirsk, Siberian Scientific Publishing house, (2009) 457. [2] S.I.Kabanichin, A.Zh.Satybaev, Fnite-difference algorithm for solving a mixed problem for the two- dimensional wave equation // Mathematical analysis and differential equation, Novosibirsk, NSU, (1987), 45-52. [3] V.G.Romanov, Stability in inverse problems, Moscow, Scientific world? (2004), 304. [4] A.Zh.Satybaev, Finite-difference regularized solution of inverse problems of hyperbolic type, Osh, Osh regional printing house,(2001), 143. [5] A.Zh.Satybaev, Direct problem of acoustics with a cord source // Materials of the International Conference, Bishkek, (2001), 75-81.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 95

If I Could Write Mathematics It Would be a Novel: A new Approach; Curiosity and What ⇔How⇔Why

O. Gercek and M.E. Gercek Department of Computer Engineering, Girne American University, Mersin 10, Turkey [email protected]

Abstract: The beauty of mathematics and geometry can be discovered by young people and children through stories. In this project, we opened a new window in the field of actual mathematical articles with an interesting, remarkable and curious writing. In this writing, we have tried to convey misunderstandings and/or misconceptions of mathematics to many people from seven to seventy years of age in a fun, enjoyable and coherent way. We aimed to bring readers together with ”If I Could Write Mathematics, It Would be a Novel”, which will consist of mathematical stories within the nature of the basic concepts and proofs of geometry and mathematics. In order to start with a sample story in this corner, we shared my daughter’s project that she had done in fifth grade with contribution of a new approach. In her project, she aimed to describe the number of π, one of the most mysterious numbers in mathematics, in a story style to her classmates and to many people from seven to seventy years of age who could understand without feeling bored or annoyed by making it fun, enjoyable, easy and clear with a different, original method. Moreover, a new approach has been presented with hints and the questions ”What⇔How⇔Why” in this story to actual mathematics story books that have not been made so far. Thus, we have demonstrated in a fun way how to learn many concepts, methods of proof and geometric drawing with a new and curious approach; hints, noticeable warnings, remarkable questions and signs in different colors. Keywords: Mathematics education, proof methods, mathematical reasoning

2010 Mathematics Subject Classiﬁcation: 97A99, 97E50, 03F03, 97E50 References: [1] A. Umay, Hacettepe Universitesi¨ E˘gitimFak¨ultesiDergisi, 24: 234–243 (2003). [2] G. Hanna, Educational Studies in Mathematics, 44: 5–23 (2000). [3] A. J. Stylianides, D. L. Ball, Journal of Mathematics Teacher Education, 11: 307-332 (2008).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 96

The Misuse of Mathematical Proof Methods: New Approaches and Suggestions

O. Gercek Department of Computer Engineering, Girne American University, Kyrenia, Mersin 10, Turkey [email protected]

Abstract: In this research, some general misuses of mathematical proof methods and interchangeably used concepts in Turkish sources, which do not correspond conceptually, were analyzed. It was found that there is not a common language in mathematical proof methods and there are inconsistencies or missing topic descriptions in Turkish literature (math textbooks, lecture books, articles etc.). The importance and necessity of mathematical proof methods was expressed and some nice general suggestions were given in some Turkish articles. However, in these articles, it has not been seen that remarkable, speciﬁc and new recommendations or methods were suggested for a better understanding of the mathematical proof methods and awareness of the misuse of proof methods for Turkish students. A survey attempting to determine how well the proof methods are known and to what extent students are aware of the misuse of proof methods was conducted. Survey questions including Proof Opinion Questionnaire, Mathematical Proof Knowledge Test were given to students in primary mathematics education and mathematics departments. Interviews were arranged with volunteer students. Questions such as; giving reasons why students picked that answer, which proof methods were used, why that method was chosen, how it can be solved with another method, whether there are similar methods to the used methods were examined. Some questions preferred to evaluate whether they know the concepts of mathematical proof methods properly or not. It is concluded that teacher canditates and academician canditates do not know mathematical proof methods properly, they either confuse the methods or misuse them. Especially, this can be seen in 2015- 2016 academic year eleventh grade advanced math textbooks, where there were mistakes and inconsistencies in mathematical proof methods. It was a gratifying development that some of these books were generally revised after 2016. In order to minimize the mistakes and inconsistencies and to understand better the mathematical proof methods, diﬀerent approaches having not been made so far will be suggested to the mathematics teachers, university academicians and ministry of education.

Keywords: Mathematics education, proof methods, mathematical reasoning

2010 Mathematics Subject Classification: 97A99, 97E50, 03F03, 97E50 References: [1] S. N. Kutlu, B. Kutlu, Modern Temel Matematik, EkoMatbaası (1978). [2] A. Demiralp, T. Pelit, M. Gürkan, Liseler I¸cin˙ Matematik 1, Ba¸sarıYayınları (1994). [3] H. H. Hacısalihlio˘glu, Lineer Cebir Cilt 1(Be¸sinci Baskı), Gazi Universitesi¨ Yayınları (1996). [4] A. Umay, Hacettepe Universitesi¨ E˘gitimFakültesiDergisi, 24, 234–243 (2003). [5] K. Altıparmak and T. Ozis, Ege E˘gitimDergisi, 6(1), 25-37 (2005). [6] B. Çelik, Soyut Matematik 1, Dora Yayınları (2010). [7] S. Basturk, Educational Studies, 36(3), 283-298 (2010). [8] G. Güler,E. Ozdemir,¨ R. Dikici, Kastamonu E˘gitimDergisi, 20(1),219-236 (2012). [9] Komisyon, 9. Sınıf Matematik, Esen Yayınları (2012). [10] Komisyon, Ortaö˘gretim 10. Sınıf Geometri, MEB Yayınları (2012). [11] M. Doruk, A. Kaplan, Journal of Research in Education and Teaching, 2(1), 231-240 (2013). [12] Alaattin Altuntas, 11. Sınıf Ileri˙ Matematik, Birey Yayınları (2015). [13] Komisyon, Ortaö˘gretim Matematik Ileri˙ Düzey11, Ipekyolu˙ Yayınları (2015). [14] Komisyon, 11. Sınıf Ileri˙ DüzeyMatematik 1. Kitap, KarekökYayınları (2017). [15] Komisyon, Ortaö˘gretim Fen Lisesi Matematik 9, MEB Yayınları (2018).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 97

A diﬀerent critical approach to the teaching and obtaining of some elementary geometric inequalities

O. Gercek Department of Computer Engineering, Girne American University, Kyrenia, Mersin 10, Turkey [email protected]

Abstract: In this research, some misuses of an elementary geometric inequality used concepts in Turkish sources were analyzed. It was found that there are inconsistencies or incorrect results for this elementary geometric inequality in Turkish resourses (university preparation course books and math textbooks, etc.). A more useful inequality by specifying the largest lower and smallest upper bounds was found, to be able to end the inaccuracy of given inequality. After searching this geometric inequality in Turkish geometry studies, it was observed that a researcher had studied in [4,5] such as this work. But, in this study, our proofs are completely different, more simple and clearer. In [5], he gave a problem related to this inequality. He insisted this problem he had given could not be solved through geometry but through calculus. In the second part of this study, solution to this problem using the geometrical way was given successfully. Lastly, a new different approach for teaching these elementary geometric inequalies would be suggested for for the first time for the mathematics teachers and writers of math books, since the erroneous questions about this geometric inequality have been prepared and solved or answered incorrectly in the most widely used resource books by students such as those given in [6-10] until this time.

References: [1] H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited, MAA, (1967). [2] D. Pedoe, Circles: A Mathematical View, MAA, (1995). [3] S¸. Bilgi¸c,Z. Kıyak, J. Gök¸sen, Lise Geometri 9, MEB Yayınları (2004). [4] M. Ya˘gcı, Matematik Dünyası, 2004(1), 58-61 (2004). [5] M. Ya˘gcı, Matematik Dünyası 2004(4), 79 (2004). [6] S¸. Ertem, F. Kola, Ç. Yazıcı, OSS¨ Geometri Konu Anlatımlı, fdd yayınları (2007). [7] Komisyon, YGS ve LYS’ye Hazırlık Geometri, U˘gurDershaneleri (2010). [8] Komisyon, YGS-LYS Geometri Konu Ozetli¨ Soru Bankası, Zafer yayınları (2011). [9] Komisyon, YGS ve LYS Geometri Konu Ozetli¨ Soru Bankası, Dahi Adam Yayıncılık (2017). [10] S. Kütük,M. N. Ozdemir,¨ TYT Geometri Soru Bankası, KarekökYayınları (2017).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 98

M. Tunay1, O. Gercek2 1 Department of Computer and Instructional Technology Teaching, Girne American University, Kyrenia, Mersin 10, Turkey [email protected] 2 Department of Computer Engineering, Girne American University, Kyrenia, Mersin 10, Turkey [email protected] Abstract: A new approach for ranking visual career counselling based on distance measures is explained. In this paper, the traditional method of career counselling will be carried out by means of distance consulting services to make career counselling. Especially, professional education in the capacity limitation and professional working demand for education has led to a proliferation of virtual applications. Hence, e-Visual Career Application has been developed as a web based program for clients who have taken professional counselling eﬃciently. With this application, it will help clients in the process of discovering and making decisions to their interests, abilities, personalities, skills and values. This paper relies on the implications of introducing online visual career consulting in the login sessions and how it aﬀects the people (supervisor and client), the processes (consulting), and the organizations involved.

Keywords: Visual Career Counselling, Traditional Method, Virtual Application, e- Visual Career ApplicationReferences:

[1] R. Paloﬀ & K. Pratt, Building Learning Communities in Cyberspace, San Francisco: Jossey-Bass. 1999. [2] C. Zirkle, Contemporary Education, 72 (2), 39-42 2001. [3] J. Flowers, Journal of Technology Education, 13 (1), 17-30 2001. [4] C. Zirkle, Journal of Vocational education Research, 28(2), 161–181 2003. [5] M. Simonson, S. Smaldino, M. Albright & S. Zvacek, Teaching and learning at a distance (2nd ed.) Upper Saddle River, NJ: Merrill Prentice Hall 2003.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 99

Mathematics through the Mathematical Modeling Method: The Geometrical Objects

Meryem G. Cumhur Department of Mathematics Education, Art and Science Faculty, Near East University, Northern Cyprus, 99138 [email protected]

Abstract: The aim of this research is to investigate the effect of education on problem-solving skills on the Mathematical Modeling for the “Geometric Objects” unit. It was found that the topics are not supported by vital examples, they are quickly forgot- ten, draw less attention from the students and so students are not interested. According to the Mathematical Modeling Method, when examples are given in lessons related to daily life and students are asked to give similar examples to these, students get encour- aged.It should not be overlooked that the preliminary knowledge of pre-instructional students is extremely important in terms of the planning of teaching activities. How- ever, many teachers working in schools do not have enough knowledge about the different methods that can be used to detect unawareness or preliminary information or misconceptions, especially about preparing geometry lesson plans with the Mathematical Modeling Method. Mathematics, especially geometry lessons, can play a key role in removing low stu- dent performance by using the Mathematical Modeling Method about problem-solving strategies, trying to solve problems, liking problem-solving, and having confidence in problem-solving, to improve students’ problem-solving skills and structure. We know that mathematics is one of the most problematic lessons in all the countries in the world. Especially when the mathematics curriculum is intensive, mathematics’ teachers need to make the right choices to produce permanent solutions when choosing the teaching model. They can use the mathematical modeling method in the classroom environment to obtain good results in the academic achievement of the students and to improve their perceptions towards mathematics. Keywords: mathematics, geometric objects, mathematical modeling, problem-solving skills 2010 Mathematics Subject Classification: 97M10, 97A30, 97D40, 97D80, 97G30. References: [1] S. E. Toksoy & A. R. Akdeniz, Determining students’ problem solving processes via hint supported problem solving instrument. Hacettepe University Journal of Education, 32(1), 185-208, 2017. [2] E. B. Guzel, Mathematical modeling in mathematics education for researchers, educators and students. PEGEM-A Publishing. Ankara (2016). [3] P. Frejd, Teachers’ conceptions of mathematical modelling at Swedish Upper Secondary school. Journal of Mathematical Modelling and Application, 1(5), 17-40, 2012. [4] A. S. Zeytun, B. Cetinkaya & A. K. Erbas, Understanding prospectives teachers’ mathematical modeling processes in the context of a mathematical modeling course. Eurasia Journal of Mathematics, Science & Technology Education, 13(3), 691-722, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 100

Other Topics (Algebra, Geometry, Topology...)

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 101

Existence and uniqueness of solutions for the ﬁrst-order nonlinear integro-diﬀerential equations with three-point boundary conditions Misir Mardanov1, Yagub Sharifov2, Farah Zeynally3, 1 Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan [email protected] 2 Baku State University, Baku, Azerbaijan [email protected] 3 Ganja State University, Ganja, Azerbaijan [email protected]

Abstract: Mathematical modeling of many processes occurring in the real world often leads to the study of the initial and boundary value problems for ordinary and partial differential equations. Such problems form the basis of mathematical physics. From this point of view integro-differential equations play an important and special role. In this article, we study existence and uniqueness of solutions of nonlinear differential equations of the type Z t (1)x ˙(t) = f (t, x(t)) + k(t, τ, x(τ)) dτ, t ∈ (0,T ) 0 with three-point boundary conditions

(2) Ax(0) + Bx(t1) + Cx (T ) = α. Here A, B, C ∈ Rn×n are constant square matrices of order n; α ∈ Rn is a given vector; det N 6= 0; N = (A + B + C),f : [0,T ] × Rn → Rn and k : [0,T ] × [0,T ] × Rn → n R are given functions, and t1 satisﬁes the condition of 0 < t1 < T . We denote by C ([0,T ] ,Rn) the Banach space of all continuous functions from [0,T ] into Rn with the norm kxk = max {|x (t)| , t ∈ [0,T ]} , where |·| is the norm in the space Rn. We prove new existence and uniqueness results by using the ﬁxed point theorems, and Banach contraction principle. Similar problems for two-point boundary value problems are considered in [1-3].

Keywords: three-point boundary conditions, integro-diﬀerential equations, existence and uniqueness of solutions

2010 Mathematics Subject Classification: 34A34, 34B10 References: [1] A. Ashyralyev, Y.A. Sharifov, Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions, Advances in Difference Equa- tions, 2013:173, 2013. [2] M. Mardanov, Y. Sharifov, H. Molaei, Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions, Electronic Journal of Differential Equations, vol. 2014, no. 259, 1–8, 2014. [3] Y.A. Sharifov, F.M. Zeynally, Existence and uniqueness of solutions to a system of nonlinear integro-differential equations with two-point boundary conditions, The 8th international conference on differential and functional differential equations, 163–164.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 102

Singularly perturbed ﬁrst-order equations in complex domains that lose their uniqueness under degeneracy Kurmanbek Alybaev1, Aitbu Murzabaeva2 1 Jalal-Abad State University,Kyrgyzstan 2 Osh Technological University, Kyrgyzstan [email protected], [email protected] Abstract: In this paper we consider a singularly perturbed ﬁrst-order equation in complex domains, the degenerate equation of which has several isolated solutions. Cases when degenerate equations have unique solutions are considered in [1-2]. Let consider the equation: (1) εz0(t, ε) = ft, z(t, ε), where ε > 0 is a small parameter, C is a set of complex numbers, Ω is a simply connected 0 domain, with the initial condition z(t0, ε) = z where t0 ∈ Ω ⊂ C and is internal point. Constants independent of ε are denoted by M0,M1,M2... For ε = 0, from (1) we get the degenerate equation (2) ft, ξ, (t)= 0,

Let the equation (2) has solutions ξj (t), j = 1, 2, , n. Definition 1. If ∀(k 6= m) ∧ ∀t ∈ Ω(|ξk(t) − ξm(t)| > M0),then ξk(t) and ξm(t) are said isolated solutions in Ω. U1. Let f(t, z) the analytic function with respect to the variables (t, z) in some domain D the variables (t, z). Definition 2. If:1.∀t ∈ Ωj ⊂ Ω there exists z(t, ε) the solution of problem (1)-(2) 2.∀t ∈ Ωj limε→0 z(t, ε) = ξj (t) then the domain Ωj are said the domain of attraction of the solution ξj (t). Theorem. Suppose that the condition U1 are satisfied. Then for each ξj (t) there exists: 0 0 1. The solution zj (t, ε) of equation(1)satisfying condition zj (t0, ε) = zj , | zj − ξj (t0) |≤ M1ε. 2. The domain Ωj ⊂ Ω and ∀t ∈ Ωj (zj (t, ε) → ξj (t)byε) Keywords: singularly perturbed equations, degenerate equation, analytical functions, isolated solutions, domain of attraction 2010 Mathematics Subject Classification: 35J05,35J08, 35J25

References: [1] K.S. Alybaev. Method of level lines for investigating singularly perturbed equations under violation of the stability condition / [Text] / .. Alybaev // Bulletin of KGNU. Series 3, Issue 6. Bishkek, 2001. 190-200.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 103

Characterzaton of some types of compactness and constructon ndex compactness ≤ τ extensons by means of unform strutures B.E. Kanetov 1, D.E. Kanetova2 1 Faculty of Mathematics and Informatics of Jusup Balasagun Kyrgyz National University, Kyrgyz Republic, 720033, Bishkek, 547 Frunze Street 2 Faculty of Pedagogik and Information Tehnologies Jalal-Abad State University, Kyrgyz Republic, 720071, Jalal-Abad, 57 Lenina Street bekbolot [email protected]

Abstract: In his work [1] M. Stone noted that one of the interesting and diﬃcult problems of General Topology is the study of all extensions of a given topological space. B. Banashevsky [2] systematized the general problems of the theory of extensions. As is known, on every Lindelof space X is its universal uniform structure U is complete. If X0 is all-density subspace of the space X and U0 is a uniform structure on X0 induced by the uniform structure of U, then the uniform space (X,U) is completion of (X0,U0). A uniform structure U0 is not necessarily a universal uniform structure, but it has the special property of pre-Lindelof, and pre-Lindelof uniform structures of the space X0 it is possible to construct all its Lindelof extensions, considering these extensions as completions of the space X0 through to pre-Lindelof uniform structures. In this work, through the uniform structures, the index compactness ≤ τ extensions is constructed, which generalizes all Lindelof extensions, with the help of uniform structures such important compact types of topological spaces as compact, µ-compact and Lindelof spaces, also study some compact and Lindelof properties of remainders of topological spaces by means of co-coverings. References: [1] M.H. Stone. Application of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 1937, no. 41, 375-481. [2] B. Banaschewski. Extensions of Topological spaces. Canad. Math. Bull. 1964., V. 7, no. 1, P. 1-22. [3] B. Kanetov, Baigazieva N. On one property of uniform spaces. In international conference ”Func- tional analysis in interdisciplinary applications” (FAIA2017), AIP conference Proceedings 1880, edited by Tynysbek Kalmenov and Makhmud Sadybekov (American Institute of Physics, Melville, NY, 2017), 030016-1 – 030016-4; doi: 10.1063/1.5000615. [4] Borubaev. Uniform spaces and uniformly continuous mappings. Frunze, ”Ilim”, 1990. (in Rus- sian).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 104

Strongly unformly paracompactness B.E. Kanetov 1, N.A. Baigazieva2 1 Faculty of Mathematics and Informatics of Jusup Balasagun Kyrgyz National University, Kyrgyz Republic, 720033, Bishkek, 547 Frunze Street 2 Institute of Mathematics of Kyrgyz Republic National Academy of Sciences, Kyrgyz Republic, 720071, Bishkek, 265 Chui Street bekbolot [email protected]

Abstract: As we know the paracompactness play an important role in the General Topology. Therefore, the finding of uniform analogues of paracompactness is an important and interesting problem in the theory of uniform spaces. Many well-known mathematicians turned to this problem and as a result different variants of uniform paracompactness of uniform spaces appeared. For example, uniform R-paracompactness in the sense of M.D. Rice [1], uniform B-paracompactness in the sense of A.A. Borubaev [2], uniform F - paracompactness in the sense of Z. Frolik [3], uniform P -paracompactness in the sense of B.A. Pasynkov [4], uniform A-paracompactness in the sense of L.V. Aparina [5], uniform I-paracompactness in the sense of D.R. Isbell [6]. In this article we show a new approach to the definition of a uniform analog of strongly paracompactness. We introduce and investigate uniform analogues of strongly paracompact and Lindelof spaces. Their connection with other properties of compactness type is studied, and the characteristics of these classes of uniform spaces are also established by means of mappings and compact Hausdorff extensions. In particular, the problem posed by A. Borubaev: what are the uniform spaces that have ω-mapping to some strongly paracompact (separable) metrizable uniform space for any (finitely additive) open covering ω? References: [1] M.D. Rice A note on uniform paracompactness. Proc. Amer. Math. Soc, vol. 62(1977), no. 2, 359–362. [2] A.A. Borubaev. Uniform topology. Bishkek, ”Ilim”, 2013. (in Russian). [3] Z. Frolik On paracompact uniform spaces. Czechoslovak Math. J., vol. 33(1983), 476–484. [4] D. Buhagiar and B. Pasynkov. On uniform paracompactness. Czechoslovak Math. J. vol. 46(1996), no. 121, 577–586. [5] L.V. Aparina. About the uniformity and proximity of Lindelof analogues. Uspehi Math. Nauk (Achievements of Mathematical Sciences), vol. 37(1982), no. 4, 141–142. (in Russian) [6] J.R. Isbell. Uniform spaces. Providence, 1964. [7] B.E. Kanetov. Some classes of uniform spaces and uniformly continuous mappings. Bishkek, Kyrgyz National University, 2013. (in Russian). [8] D.K. Musaev. Uniformly superparacompact, quite paracompact and strongly paracompact uniform spaces. Doklady Acad. Nauk Republic of Uzbekistan (Proceedings of the Uzbekistan Rep. Academy of Sciences), 1960, no. 4, 3–9. (in Russian). [9] A.V. Arhangel’skii. On the coincidence of the dimensions of and for locally bicompact groups. Doklady Acad. Nauk SSSR (Proceedings of the USSR Academy of Sciences), vol.132(1960), no. 5, 980–981. (in Russian).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 105

Existence of immovability lines of a partial mapping of Euclidean space E59 Gulbadan Matieva1, Cholponai Abdullaeva2 1 OshSU, c. Osh, Kyrgyz Republic, gulbadan [email protected] 2Kyrgyz-Uzbek University c. Osh, Kyrgyz Republic, [email protected]

Abstract: In domain Ω ⊂ E5 it is considered a set of smooth lines such that through −→ a point X ∈ Ω passed one line of given set. The moving frame < = (X, e i)(i, j, k = 1, 5 1 −→ is frame of Frenet for the line ω of the given set. Integral lines of the vector fields e i 4 −→ are formed net Σ5 of Frenet. There is exist the point F5 ∈ (X, e 5) on the tangent of 5 4 the line ω . When the point X is shifted in the domain Ω the point F5 describes it’s 4 4 4 4 4 domain Ω5 in E5. It is defined the partial mapping f5 :Ω → Ω5 such that f5 (X) = F5 . Necessary and sufficient conditions of immovability of lines

−→ −→ −→ 4 (X, e 1), (X, e 2), (X, e 3) in partial mapping f5 are proved.

Keywords: Partial mapping, cyclic net of Frenet, Frenet frame, pseudofocus, immovability of line.

References: [1] P.K. Rashevskey, Riemann geometry and tensor analysis(in Russian). Moscow. Nauka.1967.-p.481- 482. [2] Phinikov S.P. Method of exterior form of Kartan in differential geometry(in Russian). M.-L.: Gosttexizdat,. 1948. P.-432. [3] V.T. Bazylev, About many dimensional nets in Euclidean space (in Russian) // Litov. Math. Journal, 1966. Vl. 14.-p. 475-491. [4] G.Matieva, Geometry of a partial mappings, nets and distributions of Euclidean space. (in Rus- sian). Osh, 2003. [5] V.T. Bazylev, Many- dimensional nets of double Lines // in te book: Differential geometry of a manifolds of a figure, Kaliningrad, 1975. Ussue 6.-p.19-25. [6] G. Matieva, Ch.H. Abdullaeva, Necessary and sufficient conditions of degeneracy of some partial 1 mapping of the space E5 // Sciense periodical edition “IN SITU”. ISSN 2411-7161, 6 / 2016. P.5-9. [7] Ch.H. Abdullaeva, About double lines of the partial mapping f5 in Euclidean space (in Rus- sian) E5// Information as thruster of science progress. International science-practical conference.MCII “Omega science”-chelyabinsk, 2016-c. 3-7. 4 [8] Ch.H. Abdullaeva, About double lines of partial mapping f5 of Euclidean space (in Russian) E5// Actual problems in modern science and ways to solve them. XXIX-International science-practical conference. ”Eurasian Union of Scientists-Moscow”, 18 part1, 2016-c. 85-89.

Mean value of an arithmetic function associated with the Piltz divisor function Meselem Karbas 1, Abdallah Derbal2 1 Faculty of Science and Technology Djilali Bounaama Khemis Miliana University, 44225 , (Algeria) [email protected] 2 Department of Mathematics. E.N.S., BP 92. Kouba 16016, Algiers, (Algeria) [email protected]

Abstract Let k be a ﬁxed integer, we deﬁne the multiplicative function Dk (n) = dk(n) ∗ ∗ , where dk (n) is the Piltz divisor function and dk is the unitary analogue function of dk(n) dk. The main purpose of this paper is using the elementary methods to study the mean value of the function Dk (n). Keywords : The Piltz divisor function, mean value, unitary analogue function.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 106

On the Menger algebras of quasi-open multiplace maps Firudin Muradov1 1 Near East University, Department of Mathematics, TRNC ﬁ[email protected]

Abstract: Investigation of topological spaces via groups, rings, semigroups and other algebraic structures of mappings plays an important role in modern mathematics. Many researchers have focusedtheir eﬀorts on the abstract characterization of topological spaces by semigroups of continuous, open, quasi-open and closed mappings, particularly [2], [3]. Although these deal with the maps of one variable, the wider, but not less important class of multiplace maps are known to have various applications not only mathematics itself, but are also widly used in the theory of many-valued logics, cybernetics and general systems theory. Various natural operations are considered on the sets of multiplace functions. The main operation is the superposition, i.e., the operation which as a result of substitution of some functions into other intsead of their arguments gives a new function. The study of functions of n− variables was initiated by K. Menger [1] and continued by R.M. Dicker, B.M. Schein, W.A. Dudek and V.S. Trokhimenko. An (n + 1)−ary operation [ ] deﬁned on G is superassociative, if for all x, y1, ..., yn, z1, ..., zn ∈ G the following identity holds:

[[x, y1, ..., yn] z1, ..., zn] = [x, [y1, z1, ..., zn] , ..., [yn, z1, ..., zn]] An (n+1)−ary algebra (G; [ ])satisfying the above identity is called a Menger algebra of rank n. For n = 1 it is an arbitrary semigroup. A function f between topological spaces X and Y is quasi-open if for any non-empty open set U ⊂ X, the interior of f (U) in Y is non-empty. Let Q (Xn,X) denote the Menger algebra of quasi-open functions from Xn into X with composition of functions:

[fg1...gn](a1, ..., an) = f (g1 (a1, ..., an) , ..., gn (a1, ..., an)) n where a1, ..., an ∈ X, f, g1, ..., gn ∈ Q (X ,X). The purpose of this paper is to investigate Menger algebras of quasi-open maps. It is obvious that if X and Y are homeomorphic then the Menger algebras Q (Xn,X) and Q (Y n,Y ) are isomorphic. If Q (Xn,X) and Q (Y n,Y ) are isomorphic, must X and Y be homeomorphic. In general, the answer is no. In this paper, we give an abstract characterization of Menger algebras of quasi-open functions deﬁned on a certain class of topological spaces.

Keywords: Menger algebra, homeomorphism, quasi-open map, lattice equivalence

2010 Mathematics Subject Classiﬁcation: 20N15, 54C10 References: [1] K. Menger, The algebra of functions: past, present, future. Rendicionti di mathematica, 1961, 20, p.409 - 430. [2] V.Sh.Yusufov, Open mappings, U.M.N. 1986, V.41, No.6, 185-186.(Russian) [3] F.Kh.Muradov, Semigroups of Quasi-open mappings and Lattice-Equivalence, International Jour- nal of Algebra, Vol.6, 2012, no. 29, 1443 - 1447.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 107

Minisymposium: Functional analysis in interdisciplinary applications

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 108

Approximation of fractional semilinear Cauchy problem in a Banach spaces Sergey Piskarev SRCC of Lomonosov Moscow State University Russia [email protected]

Abstract: This talk is devoted to approximation of fractional Cauchy problem α (Dt u)(t) = Au(t) + f(t, u(t)), 0 < t ≤ T, 0 < α ≤ 1, (1) u(0) = u0, α in a Banach space E, where Dt is the Caputo-Dzhrbashyan derivative in time, the operator A generates exponentially bounded analytic and compact resolution family Sα(·,A) and the function f(·, ·) is smooth in both arguments. The mild solution of (1) is defined as a function u(·) ∈ C([0,T ]; E) which satisfies equation Z t 0 (2) u(t) = Sα(t, A)u + Pα(t − s, A)f(s, u(s))ds, 0 where for any x ∈ E one has Z ∞ Z ∞ α−1 α −λt α −λt λ R(λ ,A)x = e Sα(t)xdt, R(λ ,A)x = e Pα(t)xdt, Reλ > ω. 0 0 We consider semidiscrete approximation theorem on general approximation scheme under condition of compact convergence of resolvents. Throughout this note we mainly used techniques from our paper [1], but the right hand side is given by the function f(·, ·) defined in different way convenient for applications. So we consider approximation of operator family Pα(·) with weak singularity. The author was partially supported by grant of Russian Foundation for Basic Re- search 17 − 51 − 53008.

Keywords: Fractional equations, Cauchy problems, semilinear problems, general approximation scheme, compact convergence of operators, index of solution

2010 Mathematics Subject Classiﬁcation: 34G20, 49M25, 47D99, 65J15 References: [1] Ru Liu, Miao Li and Sergey Piskarev, Approximation of Semilinear Fractional Cauchy Problem, Comput. Methods Appl. Math., vol.15, no 2, 203–212, 2015.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 109

Cordoba-Cordoba type inequality on homogenous Lie groups Aidyn Kassymov1, Durvudkhan Suragan2 1Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected] 2Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected]

n Abstract: Let s ∈ [0, 1] and x ∈ R , n ≥ 2. In the work [1], authors show that the following inequality for the fractional Laplacian (1) 2f(x)(−∆)sf(x) ≥ (−∆)sf 2(x), s n 2 n where (−∆) is the fractional Laplacian, x ∈ R and f(x) ∈ C0 (R ). This inequality is using for the maximum principle of the quasi-geostrophic equations. Also, in the works [2] generalized the Cordoba-Cordoba inequality, (2) pf(x)(−∆)sf(x) ≥ (−∆)sf p(x), s n 2 n where (−∆) is the fractional Laplacian, p > 0, x ∈ R and f(x) ∈ C0 (R ). In the work [3], author generalized these inequalities for the fractional Laplacian. Our main aim of this talk is to establish analogues of the Cordoba-Cordoba inequality and its generalizations for the fractional sub-Laplacian on the homogenous Lie groups. In this talk, we show an analogue of the Cordoba-Cordoba type inequality for the fractional sub-Laplacian on the homogenous Lie groups. Also, we show generalized analogueof the Cordoba-Cordoba type inequality on the homogenous Lie groups. The authors were supported in parts by the grant AP05130981 as well as by the MES RK target programm BR05236656.

Keywords: Cordoba-Cordoba inequality, fractional sub-Laplacian, homogenous Lie groups.

2010 Mathematics Subject Classiﬁcation: 22E30, 43A80 References: [1] A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations. Com- mun. Math. Phys., 249: 511–528, (2004). [2] N. Ju. The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations.Commun. Math. Phys., 255: 161–181, (2005). [3] J. Wu. Lower bound for an integral involving fractional Laplacians and the generalized Navier- Stokes equations in Besov spaces. Commun. Math. Phys., 263: 803–831, (2006).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 110

Distribution of eigenvalues of a third-order diﬀerential operator with strongly regular boundary conditions Nurlan S. Imanbaev 1

1 South-Kazakhstan State Pedagogical University, 13 Baytursinov str., 160012 Shymkent, Kazakhstan [email protected]

Abstract: This paper is devoted to the differential operator, generated by the ordinary differential equation and nonlocal boundary conditions. It is proved that the characteristic determinant of the considered spectral problem is an entire function. The paper was supported by the state grant ”The Best Teacher of the University - 2017” of the Ministry of Education and Science of the Republic of Kazakhstan References: [1] G. D. Birkhoff, On the asymptotic chapacter of the solutions of certain linear differential equations containing a papameter, Transactions of the American Mathematical Society, Vol. 9, 2, 219–231 (1908). [2] G. D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Transactios of the American Mathematical Society, Vol. 9, 4, 373–395 (1908). [3] M. A. Naimark, Linear Differential Operators, Moscow: Science, 496 p. (1969). [4] M. A. Sadybekov and N. S. Imanbaev, Reqular Differential Operator with Perturbed Boundary Condition, Mathematical Notes, Vol. 101, 5, 878–887. (2017). [5] N. S. Imanbaev, Stability of the basis property of eigenvalue systems of Sturm - Liouville operators with integral boundary condition, Electron. J. Diff. Equ, Vol. 2016, 87, 1–8 (2016).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 111

One class of inverse problems for reconstructing the process of heat conduction from nonlocal data Makhmud Sadybekov1, Gulnar Dildabek1,2, Marina B. Ivanova1,3 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, 2 Al-Farabi Kazakh National University, Kazakhstan 3 South-Kazakhstan State Pharmaceutical Academy, Kazakhstan [email protected], [email protected], [email protected]

Abstract: We consider one class of inverse problems for a one-dimensional heat equation with involution and with periodic type boundary conditions with respect to a space variable. We will consider the process which is so slow that it is described by an heat equation. Thus, this process is described by the equation

(1) DtΦ(x, t) − Φxx (x, t) + εΦxx (−x, t) = f (x) in the domain Ω = {(x, t): −π < x < π, 0 < t < T }. Here f(x) is the inﬂuence of an external source that does not change with time; t = 0 is an initial time point and t = T is a ﬁnal one.

In this report, several variants of the operator Dt will be considered. Problems for a classical heat equation and for an equation with fractional derivatives with respect to a time variable are considered:

i) DtΦ(x, t) = Φt (x, t); −β α α 1−α d ii) DtΦ(x, t) = t D Φ(x, t) ,D ϕ (t) = I ϕ (t) , 0 < α < 1. C t C t dt α 1−α Where DC t is a Caputo derivative and I is a Riemann-Liouville fractional integral. As the additional information we take values of the initial and ﬁnal conditions of the temperature (2) Φ (x, 0) = φ (x) , Φ(x, T ) = ψ(x), x ∈ [−π, π].

Since the wire is closed, it is natural to assume that the temperature at the ends of the wire is the same at all times: (3) Φ (−π, t) = Φ (π, t) , t ∈ [0,T ] .

Consider the process in which the temperature at one end at every time point t is proportional to the rate of change speed of the average value of the temperature throughout the wire. Then, Z π (4) Φ (−π, t) = γDt Φ(ξ, t) dξ, t ∈ [0,T ] . −π Here γ is a proportionality coefficient. Thus the investigated process is reduced to the following mathematical inverse problem: Find a right-hand side f(x) of the heat equation (1), and its solution Φ(x, t) subject to the initial and final conditions (2), the boundary condition (3), and condition (4). This research is financially supported by a grant AP05133271 and by the target program BR05236656 from the Science Committee from the Ministry of Science and Education of the Republic of Kazakhstan.

Keywords: Inverse problem; heat equation; equation with involution; periodic boundary conditions; method of separation of variables

2010 Mathematics Subject Classiﬁcation: 35K20, 35R30, 35R10 References: [1] B. Ahmad, A. Alsaedi, M. Kirane, R.G. Tapdigoglu; An inverse problem for space and time fractional evolution equations with an involution perturbation, Quaestiones Mathematicae, vol. 40, no. 2, 151-160, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 112

On coersive solvability of a third-order diﬀerential equation with oscillating intermediate coeﬃcient Kordan Ospanov1, Zhuldyz Yeskabylova2 L.N. Gumiliov Eurasian National University, Kazakhstan 1 2 [email protected], juli−[email protected]

Abstract: In this report the sufficient conditions of well-posedness in L2 (−∞, +∞) of the following differential equation −y000 + p(x)y0 + q(x)y = f(x), (1) with smooth, unbounded and fast oscillating coefficients are received. We give the maximal regularity estimate 000 0 ky k2 + kpy k2 + kqyk2 ≤ Ckfk2 for the solution y (k · k2 is a norm in L2 (−∞, +∞)). Here the intermediate coefficient p is not controlled by q, and |q| is an unbounded function. The conditions obtained us are close to necessary. In the case when the domain is unbounded, although the solution of the odd-order equation (1) is smooth, but it may not belong to any Sobolev space, or its belongs to the Sobolev space is not known in advance. This is one of its main differences from the Sturm-Liouville equation. So, the smoothness problems for solutions of the equation (1) are of great interest. For such equations is important to study of separability or, in other words, the maximal regularity problem. In the case p = 0 the maximal regularity problem for the equation (1) and its quasilinear generalizations was investigated by M. Otelbaev, M.B. Muratbekov, K.N. Ospanov and etc. Moreover, in their works the existence of a solution was proved, and when L is linear, the uniqueness of the solution and the important spectral and approximate properties of the resolvent L−1 were given. The authors were supported by the grant no. AP05131649 of the Ministry of Educa- tion and Science of Republic of Kazakhstan.

Keywords: diﬀerential equation, unbounded domain, oscillating coeﬀcients, well-posedness, coercive estimate

2010 Mathematics Subject Classiﬁcation: 34A30, 34B40

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 113

On a generalization of the Neumann problem for the Laplace equation B.Kh.Turmetov 1, K.Dz.Nazarova2 1 Akhmet Yasawi International Kazakh-Turkish University, Sattarkhanov street 29, 161200 Turkestan, Kazakhstan [email protected], [email protected]

Abstract: In the paper we investigate solvability of a fractional analogue of the Neu- mann problem for the Laplace equation. As a boundary operator we consider operators of fractional differentiation in the Hadamard sense. The problem is solved by reduction to the integral Fredholm equation. Theorem on existence and uniqueness of the problem solution is proved. This research is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan,Grant No.AP05131268. References: [1] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 2006. [2] B. Kh. Turmetov, M. D. Koshanova and K. I. Usmanov, Electronic Journal of Differential Equa- tions, 2016, 1–12 (2016). [3] B. Kh.Turmetov, Diff.Equations, 53:3, 333–344 (2017).

On Green’s function of certain boundary-value problems for the biharmonic equation B.Kh.Turmetov Akhmet Yasawi International Kazakh-Turkish University, Sattarkhanov street 29, 161200 Turkestan, Kazakhstan [email protected] Abstract: In this paper we study the method of constructing the Green’s function of the Neumann and Robin problems analogues for biharmonic equation in the unit ball. Integral representations of the Green’s function of these problems are obtained. The work was supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant No. AP05131268).

References: [1] V. V. Karachik, B. Kh. Turmetov, and A. E. Bekaeva, IJPAM, 81:3, 487-495 (2012). [2] H. Begehr, and T. Vaitekhovich, Complex Var. Elliptic Equ., 58:4, 483–496 (2013). [3] V. V. Karachik, and B. Kh. Turmetov, Adv.Pure Appl. Math., 81:3, 487-495 (2018). References: [4] M. A. Sadybekov, B. T.Torebek, and B. Kh.Turmetov, AIP Conference Proceedings, 1611, 255– 260, (2014). [5] M. A. Sadybekov, B. Kh.Turmetov, and B. T.Torebek, Complex Var. Elliptic Equ., 61, 104– 123,(2016). [6] M. A. Sadybekov, B. Kh.Turmetov, and B. T.Torebek, Adv.Pure Appl. Math., 6, 102–107, (2015).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 114

On the number of eigenvalues of correct restrictions and extensions for the Laplace operator Bazarkan Biyarov1, Gulnara Abdrasheva2 and Sandugash Toleshbay3 L.N. Gumilyov Eurasian National University, Astana, Kazakhstan 1 [email protected] 2 [email protected] 3 [email protected]

Abstract: At the beginning of the last century J. Hadamard constructed the well- known example illustrating the incorrectness of the Cauchy problem for elliptic-type equations. If the Cauchy problem for some differential equations is correct, then it is usually a Volterra problem, that is, the inverse operator is a Volterra operator. For such differential equations there exists a set of Volterra correct problems besides the Cauchy problem. Until recently, not a single Volterra correct restriction or extension for elliptic-type equations was known. Biyarov (see [1]) has recently proved that there are no Volterra correct restrictions and Volterra correct extensions for the Laplace operator in the two-dimensional case, it has also proved that there are no Volterra correct restrictions and extensions for a wide class of correct restrictions and extensions for the Laplace operator in the n-dimensional case. In the present paper, we prove the theorem that the correct restrictions and extensions for the Laplace operator with discrete spectrum always have an infinite countable number of eigenvalues in the two-dimensional case.

Keywords: Laplace operator, maximal (minimal) operator, Volterra operator, correct restrictions, correct extensions, discrete spectrum, number of eigenvalues

2010 Mathematics Subject Classiﬁcation: Primary 47Axx, 47A10, 47A75; Sec- ondary 47Fxx References: [1] B. N. Biyarov, On the spectrum of well-deﬁned restrictions and extensions for the Laplace operator, Math. notes, 95, No. 4, 463–470 (2014) (Translated from mathematicheskie zametki, 95, No. 4, 507–516 (2014)).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 115

An expansion result for the equation of transverse vibration of two-layered composite Serife Faydaoglu Dokuz Eylul University, Izmir,˙ Turkey. [email protected]

Abstract: This paper is concerned with the eigenfunctions expansions of fourth- order diﬀerential equations with impulse. Applying the Fourier series approach we ﬁnd a solution of this problem and establish a uniformly convergent expansion formula in the eigenfunctions. It is necessary in applications to construct such eigenfunctions and discuss the convergence of expansions.

Keywords: Spectral problem, impulse conditions, eigenvalues, eigenfunctions

2010 Mathematics Subject Classiﬁcation: 35L05, 35P10, 47A75 References: [1] S. Faydaoglu and G.Sh. Guseinov, Eigenfunction expansion for a Sturm- Liouville boundary value problem with impulse, International Journal of Pure and Applied Mathematics, vol. 8, no 2, 137- 170, 2003. [2] S. Faydaoglu and G.Sh. Guseinov, An expansion result for a Sturm-Liouville eigenvalue problem with impulse, Turk. J. Math., vol. 34, no 3, 355-366, 2010. [3] S. Faydaoglu, On spectral properties of a fourth-order boundary value problem with impulse, 12th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM-2014), AIP Conf. Proc. 1648, 2015, Rhodes, Greece.

A tri-harmonic Neumann function for the unit disc Burgumbaeva Saule1, 1Faculty of mechanics and mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan [email protected]

Abstract: In this paper we consider a tri-harmonic Neumann function for the unit disc. The Neumann problem is well studied for harmonic functions and solved under certain conditions via the Neumann functions, sometimes also called Green function of second kind [1]. A tri-harmonic Neumann function is constructed in an explicit way for the unit disc of the complex plane by convoluting the harmonic with a bi-harmonic Neumann function. With this Neumann function an integral representation formula is developed for the try-harmonic operator. References: [1] W.Haack, W. Wendland, Lectures on partial and faffian differential equations, Pergamon Press, Oxford,, 1972. [2] H.Begehr, C.J. Vanegas, Neumann problem in complex analysis. Proc. 11th Intern. Conf. Finite Dimensional Complex Anal., Appl., Eds. P. Niamsup et al. Chiang Mai, Thailand, 2003, 212-225. [3] H.Begehr, C.J. Vanegas, Iterated Neumann problem for the higher order Poisson equation, Math. Nachr. 279 , 38-57. [4] S.Burgumbayeva, Boundary value problems for tri-harmonic functions in the unit disc, Ph.D. thesis, FU Berlin, 2009: www.diss.fu-berlin.de/diss/receive/FUDISS-thesis-000000012636. [5] Y.Wang, Boundary Value Problems for Complex Partial Differential Equations in Fan-shaped Domains, Ph.D. thesis, FU Berlin, 2010: www.diss.fu-berlin.de/diss/receive/FUDISS-thesis- 000000009061.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 116

About the Generalized Dirichlet – Neumann Problem for an Elliptic Equation of High Order Bakytbek D. Koshanov1, Alexander P. Soldatov2 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan and Kazakh National Pedagogical University named after Abai, Kazakhstan [email protected] 2 Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, Moscow, Russia [email protected]

Abstract: For the elliptic equation 2l− th order with constant (and only) real coeﬃcients considered boundary value problem of the job normal derivatives the (kj −1)− order, j = 1, . . . , l, where 1 ≤ k1 < . . . < kl ≤ 2l − 1. When kj = j it moves to the Dirichlet problem, and when kj = j + 1 - in the Neumann problem. In this paper, the study is carried out in space C2l,µ(D). The suﬃcient condition of the Fredholm tasks and present a Formula for its index. n Let s ∈ [0, 1] and x ∈ R , n ≥ 2. In the work [1], authors show that the following inequality for the fractional Laplacian (1) 2f(x)(−∆)sf(x) ≥ (−∆)sf 2(x), s n 2 n where (−∆) is the fractional Laplacian, x ∈ R and f(x) ∈ C0 (R ). This inequality is using for the maximum principle of the quasi-geostrophic equations. Also, in the works [2] generalized the Cordoba-Cordoba inequality, (2) pf(x)(−∆)sf(x) ≥ (−∆)sf p(x), s n 2 n where (−∆) is the fractional Laplacian, p > 0, x ∈ R and f(x) ∈ C0 (R ). In the work [3], author generalized these inequalities for the fractional Laplacian. Our main aim of this talk is to establish analogues of the Cordoba-Cordoba inequality and its generalizations for the fractional sub-Laplacian on the homogenous Lie groups. In this talk, we show an analogue of the Cordoba-Cordoba type inequality for the fractional sub-Laplacian on the homogenous Lie groups. Also, we show generalized analogueof the Cordoba-Cordoba type inequality on the homogenous Lie groups. The authors were supported in parts by the grant AP05130981 as well as by the MES RK target programm BR05236656.

Keywords: Cordoba-Cordoba inequality, fractional sub-Laplacian, homogenous Lie groups.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 117

On a certain class of pseudo-diﬀerential operators and related equations Vladimir Vasilyev Belgorod National Research University, Russia [email protected]

Abstract: To extend obtained results on Fredholm properties of pseudo-differential operatos on manifolds with non-smooth boundaries [1,2] we start from studying model pseudo-differential operators Z Z i(x−y)ξ m (Au)(x) = A(x, ξ)e u(y)dydξ, x ∈ R , m m R R s(x) m in local Sobolev–Slobodetskii space H (R ) with finite value  1/2 Z 2s(x) 2 ||u||s(x) ≡  (1 + |ξ|) |u˜(ξ)| dξ , m R whereu ˜ denotes the Fourier transform of the function u. Some Fredholm properties for elliptic pseudo-differential operator are obtained for a m m whole space R and a half-space R+ . Keywords: pseudo-differential operator, variable order, invertibility, Fredholm property

2010 Mathematics Subject Classification: 47G30, 35S15, 58J05 References: [1] V. B. Vasilyev, Elliptic equations, manifolds with non-smooth boundaries, and boundary value problems, In: New Trends in Analysis and Interdisciplinary Applications. Trends in Mathematics, Research Perspectives. P. Dang et al. (eds.) Birkhäuser,2017. P. 337-344. [2] V. B. Vasilyev, Pseudo-differential operators on manifolds with a singular boundary, In: Modern Problems in Applied Analysis. P. Drygas and S. Rogosin (eds.) Birkhäuser,2018. P. 169-179.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 118

Spectral Properties of the Sturm-Liouville Operator with a Parameter That Changes Sign and Their Usage to the Study of the Spectrum of Diﬀerential Operators of Mathematical Physics Belonging to Diﬀerent Types Mussakan Muratbekov1, Madi Muratbekov2 1 Taraz State Pedagogical Institute, Kazakhstan musahan [email protected] 2 Kazakh University of Economics, Finance and International Trade, Kazakhstan [email protected]

Abstract: This report is devoted to the spectral properties of the Sturm-Liouville operator with a parameter that changes sign and their application to the study of the spectral properties of differential operators of elliptic, hyperbolic and parabolic types. Consider the differential operator ∂2u ∂2u (1) Lu = k (y) − + b (y) ux + m (y) u ∂x2 ∂y2 ∞ 2 originally defined on C0 R , where k (y) is a piecewise continuous and bounded func- ∞ 2 tion changing sign in R, C0 R is the set of infinitely differentiable and compactly supported functions, the coefficients b (y) , m (y) are continuous functions in R and bounded in each compact set. It is easily verified that the operator can belong to different types, depending on the sign of the coefficient k(y). For instance, let k (y) = −1, then the operator L is of elliptic type. If however, k (y) = 1 the operator will be hyperbolic. In the case k (y) = 0, the operator L is of parabolic type. It is known that the three types of partial differential operators considered here play a special role in mathematical physics. And finally, the type of the operator L may be different at different points if the function k (y) changes sign in R. We note that such operator is called an operator of mixed type. ∞ 2 Further, let u (x, y) ∈ C0 R , then it is easy to see that after applying the Fourier transformation with respect to x the operator L takes the form: 00 2 (2) Ltuˆ = −uˆyy + −k (y) t + itb (y) + m (y) u,ˆ 1 R ∞ −itx whereu ˆ (t, y) = Fx→tu (x, y) = √ u (x, y) e dx. 2π −∞ Hence it is easy to notice that questions on the existence of the resolvent and spectral properties of the operator L transfer into problems on the existence and spectral properties of the Sturm-Liouville operator with a potential involving a parameter with changing sign. The following results: 1) the existence of the resolvent, 2) the criterion of the spectrum discreteness, 3) two-sided estimates of approximation numbers (s-numbers) are obtained for operator 2. These results are applied to the study of the spectral properties of differential operators of elliptic, parabolic and hyperbolic types. Throughout this note we mainly use techniques from our works [1-5].

Keywords: resolvent, spectrum, operator of mixed type, spectral properties, Sturm- Liouville operator with a parameter

2010 Mathematics Subject Classiﬁcation: 34B24; 34L05; 35M10; 47A10 References: [1] M.B. Muratbekov, T. Kalmenov, M.M. Muratbekov, On discreteness of the spectrum of a class of mixed type singular diﬀerential operators, Complex Variables and Elliptic Equations, 60, No. 12, 2015, 1752-1763. Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 119

[2] M.B. Muratbekov, M.M. Muratbekov, Estimates of the spectrum for a class of mixed type operators, Differential Equation, 43, No. 1, 143-146 (2007); translation from Differ. Uravn. 43, No. 1, 135-137 (2007). [3] M.B. Muratbekov, M.M. Muratbekov, K.N. Ospanov, On approximate properties of solutions of a nonlinear mixed-type equation, J. Math. Sci., New York 150, No. 6, 2521-2530 (2008); translation from Fundam. Prikl. Mat. 12, No.5, 95-107 (2006). [4] M.B. Muratbekov, M.M. Muratbekov, K.N. Ospanov, Coercive solvability of odd-order differential equations and its applications, Dokl. Math. 82, No. 3, 909-911 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 435, No. 3, 310-313 (2010). [5] M.B. Muratbekov, M.M. Muratbekov, A.M. Abylayeva, On existence of the resolvent and discreteness of the spectrum of a class of differential operators of hyperbolic type, Electronic Journal of Qualitative Theory of Differential Equations 64, 2013, pp. 1-10.

About Volterra’s nonlocal boundary value problem with displacement for the wave equation A. Shaldanbayev1, I. Orazov2, M. Shomanbayeva2 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, [email protected] 2 M.Auezov South Kazakhstan state University, Kazakhstan [email protected] 2South Kazakhstan Pedagogical University, Kazakhstan [email protected] Abstract: Recently the interest in boundary value problems of the wave equation sharply have increased due to the growing needs of Geophysics and Seismology. In this paper we show that the number of boundary value problems for the wave equation, taken inside the characteristic quadrangle are Volterra. The main results of this work are presented by following theorems:

Theorem 5.1. If α2 − β2 6= 0, then a nonlocal boundary value problem with displacement

(1) Lu = uxy(x, y) = f(x, y), (x, y) ∈ Ω;

(2) α · u(0, y) + β · u(1, 1 − y) = 0,

(3) α · u(x, 0) + β · u(1 − x, 1) = 0, is strongly solvable, and the inverse operator L−1 to the strong operator of problem (1)–(3) is completely continuous and Volterra.

Theorem 5.2. If α2 − β2 6= 0, then nonlocal boundary value problem 2 (4) Lu(x, y) = uxy(x, y) = f(x, y), f(x, y) ∈ L (Ω),

(5) αu(0, y) + βu(y, 0) = 0,

(6) αu(x, 1) + βu(1, x) = 0 is strongly solvable in space L2(Ω), and the operator L−1 inverse to the strong operator of problem (4)–(6) is completely continuous and Volterra.

This research is ﬁnancially supported by a grant AP05133271 from the Science Com- mittee from the Ministry of Science and Education of the Republic of Kazakhstan.

Keywords: Volterra operators, Goursat problem, wave equation, similarity operators

2010 Mathematics Subject Classification: 35L05, 34L10, 35L20 References: [1] T.Sh.Kal’menov, M.A. Sadybekov; The Dirichlet problem and nonlocal boundary value problems for the wave equation, Differ. Uravn., 26:1 (1990), 60–65; Differ. Equ., 26:1 (1990), 55-59.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 120

On the Spectral Properties of a Wave Operator Perturbed by a Lower-Order Term Amir Sh. Shaldanbayev 1, Manat T. Shomanbayeva 2, Aygerim A. Shaldanbayeva 3 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, [email protected] 2 M.Auezov South Kazakhstan state University, Kazakhstan [email protected] 3South Kazakhstan Pedagogical University, Kazakhstan Altima [email protected]

Abstract: The incorrectness of the minimal wave operator is well known, since zero is an infinite-to-one eigenvalue for it. As our study showed, the situation changes if the operator is perturbed by a low-order term containing the spectral parameter as a coefficient, and eventually the problem takes the form of a bundle of operators. The resulting bundle of operators is easily factorized by first-order functional-differential operators whose spectral properties are easily studied by the classical method of separation of variables. Direct application of the method of separation of variables to the original bundle of operators encounters insurmountable difficulties. In this work we consider the following spectral problem: 2 (1) uxx − uyy = −2λux + λ u,

(2) u|y=0 = 0, u|x=0 = α u|x=1 , |α| = 1. We investigated the spectral properties of task(1)-(2) and obtain the eigenvalues and corresponding eigenfunctions, which form an orthonormal basis of the space L2(Ω). The spectrum of the beam of operators (1)-(2) consists of two series of inﬁnite eigenvalues. This work was supported by the Science Committee of the Ministry of Ed- ucation and Science of the Republic of Kazakhstan, grant AP05133271.

Keywords: deviating argument, strong solvability, spectrum, beam of operators

2010 Mathematics Subject Classiﬁcation: 35L20, 35D35, 47A10 References: [1] T. Sh. Kal’menov, S. T. Akhmetova, A. Sh. Shaldanbaev; To the spectral theory of equations with deviating argument // Mathematical Journal, Almaty, 2004, Vol.4, No. 3 (13), 41–48 (In Russian).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 121

Compactness Of The Commutators Of Riesz Potential On Global Morrey-type Space Bokayev Nurzhan1, Matin Dauren1, Baituyakova Zhuldyz1 1L.N. Gumilyov Eurasian National University, Astana, Kazakhstan [email protected], [email protected], [email protected]

Abstract: In this paper we obtain the suﬃcient conditions of compactness of the w commutator for the Riesz potential [b, Iα] in global Morrey-type spaces GMpθ. Let 1 ≤ p ≤ ∞, w be a measurable non-negative function on (0, ∞). The Global w w n loc n Morrey-type space GMpθ ≡ GMpθ(R ) is deﬁned as the set of all functions f ∈ Lp (R )

with ﬁnite quasi-norm kfkGMw ≡ sup w(r) kfkL (B(x,r)) , where B(t, r) the pθ n p L (0,∞) x∈R θ ball with center at the point t and of radius r. n For the function b ∈ Lloc(R ) we denote by Mb the multiplication operator Mbf = bf, where f is a measurable function. Then the commutator for the Riesz potential Iα and the operator Mb is deﬁned by the equality [b, Iα] = MbIα − IαMb = R [b(x)−b(y)]f(y) n Cn,α n−α dy. It is said that the function b(x) ∈ L∞(R ) belongs to the n |x−y| R n 1 R space BMO( ) if kbk = sup n b(x) − b dx = sup n M(b, Q) < ∞, R ∗ Q⊂R |Q| Q Q∈R Q n 1 R where Q - cub R and bQ = |Q| f(y)dy. n R n ∞ n ∞ n We denote by VMO(R ) the BMO -closure of the space C0 (R ), where C0 (R ) ∞ n the set of all functions in C (R ) with compact support. 1 1 1 Theorem 2. Let 0 < α < n(1 − ), 1 ≤ p1 < p2 < ∞, α = n( − ), or q p1 p2 1 1 n 1 1 ≤ p1 < ∞, 1 ≤ p2 < ∞ and n( − ) < α < , α < n(1 − ), 0 < θ1 ≤ p1 p2 p1 p2 1 r θ ≤ ∞, θ ≤ 1, w ∈ Ω , w ∈ Ω , and let w (r) p2 ≤ 2 1 1 p1θ1 2 p2θ2 2 n −α (t+r) p1 L θ2(0,∞) n c kw1(r)kL for all t > 0, where c > 0 not depend on t, and b ∈ VMO(R ). In θ1(t,∞) w1 w2 addition, let the commutator [b, Iα] bounded from GM to GM . More over, let p1 θ1 p2 θ2 k[b, Iα]k w2 ≤ kbk∗ kfk w1 Then the commutator [b, Iα] is a compact operator GM GMp θ p2 θ2 1 1 w from GM w1 to GM 2 p1 θ1 p2 θ2 Keywords: Compactness, Commutators, Riesz Potential, Global Morrey-type Space

2010 Mathematics Subject Classiﬁcation: 35Q79, 35K05, 35K20

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 122

Sharp by order two-sided estimates for one class of function series Seyit Kerimkhulle1 1 L.N. Gumilyov Eurasian National University, Kazakhstan [email protected]

Abstract: In this work we consider the function series ∞ X 1 (1) J := J = , t = 0, 1, 2,..., t Qi i=t j=t(1 + rj ) Qt+T −1 with function rt ≥ 0, j=t (1 + rj ) < ∞, T = 1, 2,... . We find sharp by order two-sided estimates for the function J. We note that the functional series (1) was obtained by Kerimkhulle S. and Aitkozha Zh. in [1] as a solution of the first order linear difference equation. Also note that in [2], Lukachev M. and Shuster L. obtained find sharp by order two-sided estimates in case the integral functional (1). For a proof of main results is obtained by Otelbayev (see [3]) variational methods for localization.

Keywords: function series, upper estimate, lower estimate, sharp by order two-sided estimate

2010 Mathematics Subject Classification: 39A10, 40A05, 40A10 References: [1] S. Kerimkhulle and Zh. Aitkozha, A criterion for correct solvability of a first order difference equation, AIP Conference Proceedings 1880, 040016, doi: 10.1063/1.5000632, 2017. [2] M. Lukachev and L. Shuster, Conditions for correct solvability of a first order linear differential equation in space Lp(R) and asymptotic properties of its solutions, arXiv:math/0609752, 2006. [3] K. T. Mynbaev and M. O. Otelbaev, Weighted functional spaces and the spectrum of differential operators (in Russian), Moscow: Nauka, 1988.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 123

Degenerate operators for ordinary diﬀerential equations Magzhan Biyar Department of Mathematics, Nazarbayev University, Astana, Kazakhstan [email protected]

Abstract: We consider the spectral problem for operators generated by ordinary differential equations. A degenerate operator means an operator whose point spectrum fills the whole complex plane, or a resolvent set fills the whole complex plane. Conditions are found for the coefficients of the general differential equation for which the corresponding operator is degenerate.

Theorem: Let the operator L generated in L2(0, 1) by the differential expression (n) Pn (n−j) (j−1) l(y) = y + j=1 pj (x)y , and the boundary conditions Uj (y) = y (0) − α · (j−1) 2π(k−1) (j−1) i n α2 y (1) = 0, j = 1, 2, . . . , n, where αk = e , k = 1, 2, . . . , n, being the n-th roots of unity with α1 = 1. If the condition k pk(x) = α2 pk(1 + α2x), k = 1, 2, . . . , n, 0 ≤ x ≤ 1, is satisfied, then the operator L is degenerate, that is, the characteristic determinant n n −1 ∆L(ρ) ≡ const. Furthermore, ∆L(ρ) = 1 − α , and if α 6= 1, then L is a Volterra n operator; if α = 1, then ρ(L) = ∅ and σ(L) = C. Keywords: ordinary differential equations, degenerate operator, spectrum, resolvent set

2010 Mathematics Subject Classification: Primary 34L05 References: [1] V. A. Marchenko, Sturm-Liouville operators and applications, BirkháuserVerlag, Boston, 1986. [2] J. Locker, Eigenvalues and completeness for regular and simply irregular two-point differential operators, Amer. Math. Soc, Providence, 2008. [3] A. M. Akhtyamov, On spectrum for differential operator of odd-order, Math. notes, 101, No. 5, 755–758 (2017). [4] A. M. Akhtyamov, Degenerate boundary conditions for a third-order differential equation, Differ. Equations, 54, No. 4, 427–434 (2018) (in Russian). [5] P. Lancaster and M. Tismenetsky, The theory of matrices, Academic Press, 1985. [6] M. V. Fedoryuk, Ordinary differential equations, Nauka, Moscow, 1985 (in Russian). [7] A. M. Akhtyamov, On degenerate boundary conditions in the Sturm-Liouville problem, Differ. Equations, 52, No. 8, 1085–1087 (2016).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 124

Generalized three-dimensional singular integral equation by tube domain Nusrat Rajabov Research Institute of the Tajik National University, Tajikistan [email protected]

Abstract: In this work, we investigate one class of three- dimensional complex integral equation by tube domains, ave in lower basis and lateral surface and may have singularity. Let Ω denote the tube domain Ω = {(z, t): a < t < b, |z| < R}. Lower ground this cylinder denote by D = {t = a, |z| < R} and lateral surface denote by S = {a < t < b, |z| = R}, z = x + iy. In Ω we shall consider the following integral equation Z t K (t, τ) 1 ZZ exp[iθ]K (r, ρ) ϕ(t, z) + 1 ϕ(τ, z)dτ + 2 ϕ(t, s)ds a τ − a π (R − ρ)(s − z) D 1 Z t dτ ZZ K (t, τ; r, ρ) + 3 exp[iθ]ϕ(τ, s)ds = f(t, z), (1) π a τ − a (R − ρ)(s − z) D 2 2 2 2 2 2 where θ = arg s, s = ξ + iη, ds = dξdη, ρ = ξ + η , r = x + y , K1(t, τ) = n m P j−1 t−a P l−1 R−r Aj ln ( τ−a ), K2(r, ρ) = Bl ln ( R−ρ ), K3(t, τ; r, ρ) = K1(t, τ)K2(r, ρ), Aj (1 ≤ j=1 l=1 j ≤ n),Bl(1 ≤ l ≤ m) are given constants, f(t, z) are given function, ϕ(t, z) unknown function. In depend from the roots of the characteristics equations n m n X n−j m X m−j λ + Aj (j − 1)!λ = 0, µ + Bl(l − 1)!µ = 0 j=1 l=1 obtained representation the manifold solution of the integral equation (1), by m arbitrary functions Φl(t, z)(1 ≤ l ≤ m) analytically by variables z and continuously by variables t and n arbitrary function Cj (z) (1 ≤ j ≤ n) continuously by variables z.

Keywords: tube domain; singular kernels; manifold solution; logarithmic singularity.

2010 Mathematics Subject Classiﬁcation: 44A15, 35C10, 45E10 References: [1] N. Rajabov; New methods for Volterra type integral equation with boundary singular point, New Trends in Analysis and interdisciplinary applications, selected contributions of the 10th ISAAC congress, Macau 2015, Birkhauser, Springer International Publishing AG, 121–127, 2017. [2] N. Rajabov; Volterra type integral equation with boundary and interior ﬁxed singularity and super- singularity kernels and their application, LAP LAMBERT Academic Publishing, Germany, 2011.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 125

On a boundary condition of Bitsadze-Samarskii for the Lavrent’ev-Bitsadze equation Tynysbek Kal’menov1, Gaukhar Arepova2 Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected] [email protected]

Abstract: This report is devoted to study the Bitsadze-Samarskii boundary condition for the Lavrent’ev-Bitsadze equation. In the domain D we consider the two- dimensional integral potential: −1 − − −1 ∆N f (x1, x2), (x1, x2) ∈ D , u = LB f = −1 + −1 −1 + f (x1, x2) + 0 τ(x1, x2) + 1 ν(x1, x2), (x1, x2) ∈ D , where Z −1 − − − ∆N f (x1, x2) = ε2 (x1 − ξ1, x2 − ξ2)f (ξ1, ξ2)dξ1dξ2 (1) D− the two-dimensional Newton potential and the wave potential: Z b Z 1 −1 + + + f (x1, x2) = dξ2 ε2 (x1 − ξ1, x2 − ξ2)f (ξ1, ξ2)dξ1 (2) 0 0 and surface wave potentials: −1 + (/) uτ (x1, x2) = 0 τ(x1, x2) = ε2 ∗ τ(x1)δ (x2), (3) −1 + uν (x1, x2) = 1 ν(x1, x2) = ε2 ∗ ν(x1)δ(x2). (4) 1+α −1 1+α 2+α − When f(x1, x2) ∈ C (D) the integral operator LB f(x1, x2) ∈ C (D)∩C (D )∩ C2+α(D+) satisﬁes the equation of mixed (elliptic-hyperbolic) type:

LB u(x1, x2) = sgnx2ux1x1 (x1, x2) − ux2x2 (x1, x2) = f(x1, x2), (x1, x2) ∈ D, (5) which is called the Lavrent’ev-Bitsadze equation. In this paper, by using techniques from our works [1,2] we found the boundary conditions for the Bitsadze integral operator −1 LB f. Keywords: Newton potential, wave potential, surface wave potentials, Bitsadze-Samarskii boundary condition, Lavrent’ev-Bitsadze equation

2010 Mathematics Subject Classiﬁcation: 35J05, 35J25, 35L20 References: [1] Kal’menov T.Sh., Suragan D. On spectral problems for the volume potential. Doklady Mathematics. 80:2 (2009), pp. 646–649. DOI: 10.1134/S1064562409050032 [2] Kal’menov T.Sh., Otelbaev M., Arepova G.D. BitsadzeSamarskii Boundary Condition for Elliptic- Parabolic Volume Potential. Doklady Mahematics. 480:2 (2018), pp. 141–144.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 126

Numerical Solution of a Two Dimensional Elliptic-Parabolic Equation with Dirichlet-Neumann Condition

Allaberen Ashyralyev1, Okan Gercek2, Emel Zusi3 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey Peoples’ Friendship University of Russia (RUDN University), Ul Miklukho Maklaya 6, Moscow 117198, Russia Institute of Mathematics and Mathematical Modeling, 050010, Almaty,Kazakhstan [email protected] 2 Department of Computer Engineering, Girne American University, Kyrenia, TRNC, Mersin 10, Turkey [email protected] 3 Department of Mathematics, Luigj Gurakuqi University, Shkoder, Albania [email protected]

Abstract: In the present paper, a two dimensional elliptic-parabolic equation with Dirichlet-Neumann boundary condition is studied. The first and second order of accuracy difference schemes for the numerical solution of this problem are presented. Illus- trative numerical results of these difference schemes are provided by using a procedure of modified Gauss elimination method.

Keywords: Elliptic-Parabolic equation, nonlocal boundary value problems, stability, diﬀerence scheme

2010 Mathematics Subject Classiﬁcation: 35J15, 35K10, 35M12, 65M12 References: [1] M.S. Salakhitdinov, Equations of Mixed-Composite Type, Fan, Tashkent, Uzbekistan, 1974. [2] A.A. Samarskii, and E. S. Nikolaev, Numerical Methods for Grid Equations 2, Iterative Methods, Birkhauser, Basel, Switzerland, 1989. [3] D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylym, Ashgabat,1995 (Russian). [4] A. Ashyralyev and O. Gercek, in Further Progress in Analysis, Proceedings of the 6th International ISAAC Congress, 2009, pp. 663-670. [5] J.I. Diaz, M.B. Lerena and J.F. Padial, J.M. Rakotoson, Journal of Diﬀerential Equations 198(2), 321-355 (2004). [6] O. Gercek, and E. Zusi, AIP Conference Proceedings, 1676, 020049 (2015).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 127

On the unique solvability of a nonlocal problem with integral condition for the impulsive partial diﬀerential equation third order

Anar T. Assanova Institute of Mathematics and Mathematical Modeling, Kazakhstan E-mail: [email protected]; [email protected]

Abstract: The nonlocal problem with integral condition for the impulsive partial diﬀerential equation third order is investigated. Conditions of the unique solvability to nonlocal problem are established in the terms of initial data. In this report, on Ω = [0,T ] × [0, ω] the nonlocal problem with integral condition for the impulsive partial diﬀerential equation is considered

utxx = A(t, x)uxx + B(t, x)utx + C(t, x)ux + f(t, x), t 6= ti, (1)

Pi(x)u(ti + 0, x) − Si(x)u(ti − 0, x) = ϕi(x), i = 1, m, x ∈ [0, ω], (2)

k t L(x)u (0, x) + P R j M (t, x)u (t, x)dt = ϕ (x), x ∈ [0, ω], (3) xx tj−1 j xx 0 j=1

u(t, 0) = ψ1(t), ux(t, x) x=0 = ψ2(t), t ∈ [0,T ], (4) where u(t, x) is unknown function, the functions A(t, x),B(t, x), C(t, x), and f(t, x) are continuous on Ω, the functions Pi(x), Si(x), ϕi(x) are twice continuously differentiable on [0, ω], i = 1, m, 0 < t1 < t2 < ... < tm ≤ T , the functions L(x), ϕ0(x), Mj (t, x) are continuous on [0, ω], Ω, respectively, j = 1, k, the functions ψ1(t) and ψ2(t) are continuously differentiable on [0,T ]. Problem (1)–(4) is reduced to equivalent problem, consisting of nonlocal problem with integral condition for impulsive hyperbolic equation second order and functional relations. Algorithm of finding approximate solution to researching problem is constructed and the its convergence is proved. The conditions of classical solvability to problem (1)–(4) are established by method of introduction additional functional parameters [1, 2].

Keywords: partial diﬀerential equation, nonlocal problem, impulse eﬀect

2010 Mathematics Subject Classiﬁcation: 35G16, 35L20, 35R12 References: [1] Asanova A.T., Dzhumabaev D.S., “Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations”, J. of Math. Anal. and Appl., Vol.402, No.1, pp. 167-178, 2013. [2] Asanova A.T., “Well-posed solvability of a nonlocal boundary-value problems problem for the systems hyperbolic equations with impulse eﬀects”, Ukrainian Mathematical Journal, Vol.67, No.3, pp. 333-346, 2015.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 128

On the spectral properties of a diﬀerential operator arising in the theory of Brownian motion Kordan Ospanov L.N. Gumiliov Eurasian National University, Kazakhstan [email protected]

Abstract: It is well known that in stochastic processes is widely used the following generalized Ornstein-Uhlenbeck operator Au = −∆u + ∇u · b(x) + c(x)u, where x ∈ Rn. So, A is the generator of the transition semigroup of the stochastic process, which determines the n-dimensional Brownian motion with identity covariant matrix. The vector-valued function b is called a drift. If b = 0, then A is the Schrodinger operator, which has been systematically studied for a long time in connection with applications in quantum mechanics. If b 6= 0 and it is unbounded, then the properties of A are significally differ. For example, the well-posedness and spectral properties of A depend on the relationship between the growth at infinity of |b| and the norm of c. For more details, we refer to papers of A. Lunardi and V. Vespri (1997), G. Metafune (2001) and P.J. Rabier (2005). In this work we discuss the discreteness spectrum criterion of an folloving one- dimentional operator Ly = −y00 + r(x)y0 + q(x)y, given in Lp(R) (1 < p < ∞), where r is a continuously differentiable function, and q is a continuous function, and the growth of r not depends on q. To get the desired result, we first give the conditions for invertibility of L and describe its domain D(L). This work was supported by the grant no. AP05131649 of the Ministry of Education and Science of Republic of Kazakhstan.

Keywords: diﬀerential operator, unbounded coeﬃcients, well-posedness, discreteness of spectrum

2010 Mathematics Subject Classiﬁcation: 34L05

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 129

Duhamel principle for the time-fractional diﬀusion equation in unbounded domain Meiirkhan Borikhanov Al–Farabi Kazakh National University, Kazakhstan and Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected]

Abstract: In this paper we establish a fractional Duhamel principle for the time- fractional diﬀusion equation 2 ∂ 1−α (1) ut (x, t) − D u (x, t) = f(x, t), 0 < α < 1, x ∈ R, t > 0, ∂x2 t with the initial condition

(2) u (x, 0) = u0 (x) , x ∈ R, p where u0(x) ∈ L (R), p ≥ 1, f(x, t) is a continuously differentiable function and f(x, 0) = 0 and D1−α represents the following Riemann-Liouville fractional derivative of order 1 − α t 1 ∂ Z Dαu (x, t) = (t − s)−α f (s)ds. t Γ (1 − α) ∂t 0 In [1,2] generalized the classical Duhamel principle for the Cauchy problem to general inhomogeneous fractional distributed differential-operator equations. Keywords: Duhamel Principle, diffusion equation, fractional derivative, Green’s function.

2010 Mathematics Subject Classiﬁcation: 31A30, 31B30, 35J40.

Acknowledgments This research is ﬁnancially supported by a grant No. AP05131756 from the Ministry of Science and Education of the Republic of Kazakhstan. References: [1] S. Umarov, E. Saydamatov. A Fractional Analog of the Duhamel Principle. Fract. Calc. Appl. Anal., V. 9, No. 1, P. 57-70. 2006. [2] S. Umarov. On fractional Duhamels principle and its applications. Journal of Diﬀerential Equa- tions, V. 252, P. 5217-5234. 2012.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 130

Criterion for the unconditional basicity of the root functions related to the second-order diﬀerential operator with involution

A.M. Sarsenbi1, L.V. Kritskov2 1 M. Auezov South Kazakhstan State University, Shymkent, Kazakhstan 2 M. Lomonosov Moscow State University, Moscow, Russia [email protected] Let L be any operator related to the operation of the form 00 00 Lu ≡ −u (x) + αu (−x) + q (x) u (x) + qv (x) u (v (x)) , −1 < x < 1, (1) and defined on a dense in L2 (−1, 1) domain D (L) . The operation (1) contains the argument’s transform v0 (x) = −x, in its main term. This transform is called a simple involution (reflection) of the segment [−1; 1] and also n arbitrary involution v (x) in the lower term. The parameter α in (1) belongs to (-1;1), the coefficients q (x) and qv (x) are arbitrary and complex-valued integrable on [−1; 1] functions, the involution ν (x) is any absolutely continuous function which has an essentially bounded derivative on [−1; 1]. The particular form of the domain D (L) will not be refined below; the operator L can be generated by the functional-differential operation (3.1), for example, with some boundary conditions on the segment [−1; 1]. We only assume that the domain D (L) contains only functions that, together with their first derivatives, are absolutely continuous on the interval (−1; 1), while the root functions of the operator L are considered as regular solutions of the corresponding equations with a spectral parameter. Following Il’in [1], an eigenfunction (or a root function of the zero order) u (x) , that corresponds to the operator (3.1) and an eigenvalue λ ∈ C is defined as an arbitrary non trivial solution of the equation Lu = λu. Here and throughout, a regular solution of the equation Lu = f with a given right-hand side f ∈ L1 (−1, 1) is understood to 2 be an arbitrary function u (x) from the class W1 (−1, 1) ∩ L2 (−1, 1) , that satisfies this equation almost everywhere on (−1; 1). Letu ˜ (x) − be a root function of order (k − 1) (k ≥ 1) , corresponding to an eigenvalue λ. Then the regular solution of the equation Lu = λu − u˜ will be called its counterpart root (associated) function of order k. For each eigenvalue λ ∈ C, we have there by defined a chain of root functions uk (x; λ) , k ≥ 0 that satisfy the

Luk (x; λ) = λuk (x; λ) − sgnk · uk−1 (x; λ) , (2) moreover, u0 (x; λ) 6≡ 0 on (−1; 1) . Any count able set Λ = {λ} ⊂ C deﬁnes the system of root functions

U = {uk (x; λ) |k = 0, ..., m (λ) , λ ∈ Λ } ; here then on negative integer m (λ) will be called the rank of the corresponding eigenfunction u0 (x; λ) . Left the system U satisfy the following conditions A:

A1) the system U is complete and minimal in L2 (−1, 1) ; A2) a system V that is biorthogonally adjoint to U consists of root functions ∗ ∗ ∗ ∗ υl (x; λ ) , l = 0, ..., m (λ ) , λ ∈ Λ, m (λ ) = m (λ) , (in the above-deﬁned sense) of the formal adjoint operation

∗ 00 00 0 L υ = −υ (x) + αυ (−x) + q (x) υ (x) − v (x) qv (v (x))υ (v (x)) , (3) ∗ ∗ and the relation uk (·; λ) , υm(λ)−l (·; λ ) = 1 is valid if and only if k = l and λ = λ; while in the remaining cases the inner product on the left-hand side in relation (3.4) is zero; A3) the ranks of the eigenfunctions are uniformly bounded: sup m (λ) < ∞ λ∈Λ and the condition that the set Λ belongs to the Carleman parabole is satisfied √ sup Im λ < ∞; λ∈Λ Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 131 A4) the following uniform estimate of the ”sum of units” is valid: X sup 1 < ∞. β≥1 √ λ∈Λ:|Re λ−β|≤1 Theorem 1. Let the conditions 1-4 be satisfied and let the involution ν (x) occurring in (1) be an arbitrary continuous function with the derivative that is essentially bounded on the segment [−1, 1] . Then each of the systems U and V of root functions of the operators (1) and (3), respectively, forms an unconditional basis in L2 (−1, 1) if and only if the uniform estimate of the product of norms ku (·; λ)k · υ ·; λ¯ ≤ Mholds for k 2 m(λ)−k 2 all k = 0,..., m (λ)and λ ∈ ΛThe main theorem is complemented with the proof of the necessity of condition A4 in the case where the involution ν (x)in the operator (1) is a reflection.

Theorem 2. Let the condition A3 be satisﬁed and, in addition, let ν (x) = −x If the system of root functions U that is normed in L2 (−1, 1) possesses the Bessel property, then the uniform estimate of the ”sum of units” A4 is valid. This work was supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan, grant AP05131225. References: [1] V.A. Il’in, L. Kritskov; Properties of spectral expansions corresponding to non self - adjoint diﬀerential operators, J. Math. Sci. (NY), 116, No.5, pp. 3489-3550., 2003.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 132

Compactness results for the resolvent of a singular second order hyperbolic operator with a mixed derivative Myrzagali Ospanov L.N. Gumilyov Eurasian National University, Kazakhstan [email protected] Abstract: The resolvent compactness question for differential operators defined in an unbounded domain was studied, mainly, in the one-dimensional case, or in the case of an elliptic operator. There are works [1] and [2] devoted to the discreteness of the spectrum and other important properties of the resolvent of hyperbolic and mixed type operators. However, in these papers were considered Aˆ operators given in the Hilbert and Lebesgue spaces. At the same time, a number of practical problems lead to singular differential operators in the class of continuous functions. n Let Ω = [0; ω] × (−∞; ∞). We denote by C∗(Ω,R ) the space with norm kUk = sup |U(x, t)| (x,t)∈Ω of bounded functions U : Ω → Rn that are continuous on t ∈ R for any fixed x ∈ [0, ω] and uniformly with respect to t ∈ R continuous on x ∈ [0, ω]. n In this work in C∗(Ω,R ) we consider the following hiperbolic operator: ∂2U ∂U ∂U LU = − A(x, t) − B(x, t) − C(x, t)U. ∂t∂x ∂x ∂t We present conditions for the bounded invertibility of L and the compactness of the inverse operator L−l. Here A(x, t),B(x, t),C(x, t) are the continuous (n×n)− matrixes.

Keywords: singular hiperbolic operator, space of bounded functions, well-posedness, resolvent, compactness.

2010 Mathematics Subject Classification: 35L81, 47F05, 47A10 References: [1] M.B. Muratbekov, M.M. Muratbekov, A.M. Abylayeva, On existence of the resolvent and discreteness of the spectrum of a class of differential operators of hyperbolic type, Electronic Journal of Qualitative Theory of Differential Equations, no. 64, 1–10, 2013. [2] M.B. Muratbekov, T. Kalmenov, M.M. Muratbekov, On discreteness of the spectrum of a class of mixed-type singular differential operators, Complex Variables and Elliptic Equations, vol. 60, no. 12, 1752–1763, 2015.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 133

Structure of essential spectra and discrete spectrum of the energy operator of ﬁve-electron systems in the Hubbard model. Fifth doublet state Sadulla Tashpulatov1 1 INP of Academy of Sciences of Republik of Uzbekistan, Tashkent [email protected], [email protected], [email protected]

Abstract: We consider the energy operator of ﬁve-electron systems in the Hubbard model and describe the structure of essential spectrum and discrete spectrum of the system in the ﬁfth doublet state. The Hamiltonian of the chosen model has the form X + X + X + + H = A am,γ am,γ + B am,γ am+τ,γ + U am,↑am,↑am,↓am,↓. m,γ m,τ,γ m

Here, A is the electron energy at a lattice site, B is the transfer integral between neighboring sites (we assume that B > 0 for convenience), τ = ±ej , j = 1, 2, ..., ν, where ej are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors, U is the parameter of the on-site Coulomb interaction of two electrons, + γ is the spin index,γ =↑ or γ =↓, and am,γ and am,γ are the respective electron creation ν 5 d and annihilation operators at a site m ∈ Z . Let He1/2 denote the operator of fifth doublet state of five-electron systems and W and m are the positive numbers. 4B 4 3B 4 Theorem. Let ν = 3. a). If U < − m , m < 3 W, or U < − W , m > 3 W, then 5 d the essential spectrum of operator He1/2 is exactly the union of seven segments and the 5 d discrete spectrum of He1/2 consists of no more one point. 3B 4B 5 d b). If − W ≤ U < − m , then the essential spectrum of He1/2 is the union of four 5 d segments and the discrete spectrum of He1/2 is empty. 4B 4 3B 3B 4 c). If − m ≤ U < 0, m > 3 W, or − W ≤ U < − m , and m < 3 W, then the 5 d essential spectrum of He1/2 is the union of two segments and the discrete spectrum of 5 d He1/2 is empty. 4B 4 3B 4 d). If − m ≤ U < 0, m > 3 W, or − W ≤ U < 0, m < 3 W, then the essential 5 d spectrum of operator He1/2 is the single segment and the discrete spectrum of operator 5 d He1/2 is empty. Keywords: Hubbard Model, Essential and Discrete spectrum, five-electron systems, sextet state, quartet state, doublet states,bound and antibound states.

2010 Mathematics Subject Classiﬁcation: 47Axx, 46XX, 46L60,11F72, 35Pxx,62M15,58C40

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 134

Reduction of Navier-Stokes Equation to a Linear Equation Waleed S. Khedr Independent Researcher, Cairo, Egypt [email protected]

Abstract: In this article, we provide two theorems on pointwise coincidence between solutions of Navier-Stokes equation and solutions of standard linear second order parabolic equations with the same data. We show that the convection, the pressure, and the external forces (if applied) are governed by some sort of balance independent of the equation that governs the solution itself. In light of the well establishment of the theory of existence, regularity and uniqueness of linear second order parabolic equations, this result represents an important step to fully understand the qualitative properties of the solutions to Navier-Stokes equation.

Details: The initial profile is v0 and the boundary conditions (in case of bounded domains) are denoted by v∗. The model equation in hand is:  v + (v · ∇)v − µ∆v = −∇p + f, ∇ · v = 0 in Ω ,  t t (1) v(x, 0) = v0(x) in Ω0, ∗  v(x, t) = v (xn−1, t) on ∂Ωt, where Ωt and ∂Ωt denotes the fixed boundary if the domain is bounded. The solution v is the vector field representing the velocity of the flow in each direction, and its rotation = ∇ × v is the vorticity. Note that ∇· = 0 in Ωt by compatibility. The solution is investigated in light of the most common classical definition of weak solutions in order to ensure the possibility of generalizing the subsequent results to the widest classes of possible solutions. In particular, we follow the definition of weak solutions introduced first by Leray [1–3]. Further, the investigation in this article establishes rigorously the previously investigated results in [4, 5].

Keywords: Fluid Mechanics, Navier-Stokes equation, Convection 2010 Mathematics Subject Classiﬁcation: 76D03, 76D05, 76M30, 76R10 Acknowledgement: This is an extended abstract of the article submitted to arXiv on 24th of January 2018 with submission number arXiv:submit/2142016.

References: [1] J. Leray, Etude´ de diverses équationsintégralesnon linéaireset de quelques problémesque pose lâTMhydrodynamique, J. Math. Pures Appl., 12, 1933, 1-82. [2] J. Leray, Essai sur les mouvements plans d’un liquide visqueux que limitent des parois, J. Math. Pures Appl., 13, 1934, 341-418. [3] J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 1934, 193-248. [4] W. S. Khedr, Classical fundamental unique solution for the incompressible Navier-Stokes equation in RN , Jamp, 5, 939-952, 2017. [5] W. S. Khedr, Nonconvection and uniqueness in Navier-Stokes equation, arXiv:1706.02552, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 135

Trigonometric system and optimal approximation Fahd Jarad 1, Alexander Kushpel1, Kenan Ta¸s1 1 Department of Mathematics, C¸ankaya University [email protected], [email protected], [email protected]

Abstract: n-Widths were introduced to compare and classify the power of approximation of a wide range of algorithms. Optimality of the trigonometric system is a frequently discussed topic in the theory of n-widths [1,2]. We present a new phenomenon in the behavior of the trigonometric system in the the ”usual” order, i.e. {1, exp (±i·) , exp (±2i·) , · · ·}. Namely, the sequence of subspaces Tn of trigonometric polynomials of order n is optimal in the sense of order of Kolmogorov’s n-widths dn (K ∗ Up,Lq) on convolution classes K ∗ Up in Lq for all 1 < p, q < ∞ in the case of ”superhigh” smoothness (analytic and entire), that is in the case ∞ X K (x) = exp (−µk%) cos kx, µ > 0,% ≥ 1 k=1 [4,5] and ”supersmall” smoothness, i.e. if ∞ X − 1 − 1 K (x) = k p q + (log k)−ν cos kx, ν > 0. k=1 However, in the intermediate cases of ”small smoothness” (1/p − 1/q < r < 1/p) and ”ﬁnite smoothness” [3] (r > 1/p), i.e. if ∞ X K (x) = k−r cos kx k=1 the sequence of subspaces Tn is not optimal for all 1 < p, q < ∞. Also, we present similar results for linear and Bernstein’s n-width. References: [1] Ismagilov R. S. (1977) n-Widths of sets in linear normed spaces and approximation of functions by trigonometric polynomials, Uspekhi Mat. Nauk., 29, 161–178. [2] Kashin, B. S. (1977) n-Widths of some ﬁnite dimensional sets and classes of smooth functions, Izvestiya Acad, Nayk. USSR, 41, 334–351. [3] Kashin, B. S. (1981) On n-widths of Sobolev’s classes of small smoothness, Vestnik MGU, 5, 334–351. [4] Kushpel, A. (1993), Estimates of the Bernstein widths and their analogs, Ukrain. Math. Journ., 45, 59–65. [5] Kushpel, A. Levesley, J., Wilderotter, K. (1998), On the asymptotically optimal rate of approximation of multiplier operators, Constructive Approximation, 14, 169–185.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 136

Minisymposium: Modern Problems of Mathematical Hydrodynamics

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 137

Weak solvability of some fractional viscoelasticity models

1 Laboratory of mathematical ﬂuid dynamics of Research Institute of Mathematics, Voronezh State University, Russia orlov [email protected]

Abstract: The mathematical models of dynamics of viscoelastic ﬂuids with consti- tutive relations of the form n s m r X X k+βki X X k+βki bkiD0t σ = akiD0t ε, 0 ≤ βk < 1, (l2) k=0 i=0 k=0 i=0 α where D0t is the RiemannLiouville fractional derivative of order α > 0, σ is the deviator of the stress tensor, and ε is the deformation tensor are under consideration. Fractional analogous of Voigt and anti-Zener models are under investigation. We establish the existence of weak solutions of the corresponding initial-boundary value problems. In the planar case the uniqueness of weak solutions is proved. For the proofs of the main results we approximate the problems under consideration by a sequence of regularized systems of Navier-Stokes type. The solvability of regularized systems and apriori estimates of their solutions allow to pass to the limit in the regularized systems and obtain the solvability of original problems. The theory of fractional powers of positive operators, fractional calculus and classical results on Navier-Stokes equations are used. This is a joint work with Victor Zvyagin. This research was supported by the Ministry of Education and Science of the Russian Federation (grant 16 14.Z50.31.0037).

Keywords: viscoelastic ﬂuid; fractional derivative, Voigt model, anti-Zener model, weak solution, initial-boundary value problems.

2010 Mathematics Subject Classiﬁcation: 35A01, 35Q35, 45K05, 76D03 References: [1] V.P. Orlov , D.A. Rode and M.A. Pliev, Weak solvability of the generalized Voigt viscoelasticity model, Siberian Mathematical Journal, Vol. 58, No. 5, 859-874, 2017. [2] V.G. Zvyagin and V.P. Orlov, Weak Solvability of a Fractional Voigt Viscoelasticity Model, Dok- lady Mathematics, Vol. 96, No. 2, 1-3, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 138

Investigation of the solvability of initial-boundary value problems for a viscoelastic model with memory

V.G. Zvyagin1 1 Voronezh State University, Voronezh, Russia zvg [email protected] This is join work with Prof. V.P. Orlov.

n Abstract: Let QT = [0,T ] × Ω, where T > 0, Ω ⊂ R , n = 2, 3, is a bounded 2 domain with boundary ∂Ω ⊂ C . We consider in QT the problem n t ∂v X ∂v Z s−t (1) + vi − µ0∆v − µ1Div e λ E(v)(s, z(s; t, x)) ds + ∇p = f; ∂t ∂x i=1 i 0 Z τ (2) z(τ; t, x) = x + v (s, z(s; t, x)) ds, div v(t, x) = 0; t (3) v(0, x) = v0(x), x ∈ Ω; v(t, x) = 0, (t, x) ∈ [0,T ] × ∂Ω.

−1 Theorem 1 [1]. Let f = f1 + f2, where f1 ∈ L1(0,T ; H), f2 ∈ L2(0,T ; V ) and v0 ∈ H. Then there exists a weak solution of problem (1)-(3). n Let Q = (−∞,T ] × Ω, where T > 0, Ω ⊂ R , n = 2, 3, is a bounded domain with boundary ∂Ω ⊂ C2. We consider in Q the problem n t ∂v X ∂v Z s−t (4) + vi − µ0∆v − µ1Div e λ E(v)(s, z(s; t, x))ds + ∇p = f; ∂t ∂x i=1 i −∞ (5) div v(t, x) = 0, (t, x) ∈ Q; v(t, x) = 0, (t, x) ∈ (−∞,T ] × ∂Ω; Z τ (6) z(τ; t, x) = x + v(s, z(s; t, x)) ds, t, τ ∈ (−∞,T ], x ∈ Ω. t

−1 Theorem 2 [2]. Let f ∈ L2(−∞,T ; V ). Then problem (4)-(6) has at least one weak solution.

Keywords: Weak solution, viscoelasticity, model with memory

2010 Mathematics Subject Classiﬁcation: 35Q35, 76A10, 35D30, 35R09 This research was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037)

References: [1] V.G. Zvyagin, V.P. Orlov, Solvability of one non-Newtonian fluid dynamics model with memory, Nonlinear Analysis, vol. 172, 73–98, 2018. [2] V.G. Zvyagin, V.P. Orlov, Problem of viscoelastic fluid with memory motion on an infinite time interval, Discrete Contin. Dyn. Syst. Ser. B, 2018.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 139

Trajectory and global attractors for termo-Voigt model

Andrey Zvyagin Voronezh State University, Voronezh, Russian Federation [email protected]

n Abstract: Let Ω be bounded domain in the space R , n = 2, 3, with a smooth boundary ∂Ω. We consider the following initial-boundary value problem n ∂v X ∂v ∂∆v (1) + vi − 2Div(ν(θ)E(v)) − κ + ∇p = f; ∂t ∂x ∂t i=1 i

(2) div v = 0; v|t=0 = v0, x ∈ Ω; v|∂Ω×[0,+∞] = 0;

n ∂θ X ∂θ ∂E(v) (3) + vi − χ∆ θ = 2ν(θ)E(v): E(v) + 2κ : E(v) + g; ∂t ∂x ∂t i=1 i

(4) θ|t=0 = θ0, x ∈ Ω; θ|∂Ω×[0,+∞] = 0.

Here, v = (v1(t, x), . . . , vn(t, x)) is un unknown vector-valued velocity function of particles in the ﬂuid, p = p(t, x) is un unknown pressure, f = f(t, x) is the external force. The divergence Div C of the tensor C = (cij (t, x)) is the vector with with coordinates Pn (Div C)j = i=1(∂cij /∂xi); 1 ∂vi ∂vj E(v) = (Eij (v)), Eij (v) = + , i, j = 1, n, 2 ∂xj ∂xi is the strain-rate tensor; ν > 0 is the ﬂuid viscosity, κ is the retardation time, v0 and θ0 are given functions.

Theorem 6.1. The trajectory space H+ of problem (1)-(4) has the minimal trajectory attractor U and the global trajectory attractor A.

This research was supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037).

Keywords: Non-Newtonian ﬂuid, trajectory attractor, global attractor, existence theorem

2010 Mathematics Subject Classiﬁcation: 35B41, 35Q35, 76D03 References: [1] A.V. Zvyagin, Attractors for a model of polymer motion with objective derivative in the rheological relation // Doklady Mathematics, vol. 88, no 3, 730-733, 2013.

References: [1] P. Dittrich, The average magnetic ﬁeld in an accidental ﬂoat- Bulletin of Moscow University. Series 1. Mathematics. Mechanics, no 5, 79–82, 1984.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 140

Average magnetic field in a random stream with an update Nursadyk Akanbay 1,1, Zoya Suleimenova 1,2 1 Al-Farabi Kazakh National University, Kazakhstan 1 [email protected] 2 [email protected] Abstract: In this paper we derive the equation of the average magnetic field in a random flow with an update and an asymptotic analysis of the obtained equation is carried out. −→ −→ −→ Consider the equation of magnetic induction H = H (t, x) in a random flow V (t, x) 3 of an incompressible medium with a constant magnetic viscosity νm (t ≥ 0, x ∈ R ): −→ ∂H −→ −→ −→ −→ −→ −→ −→ (1) = νm4H − (V, ∇)H + (H, ∇)V, H (0, x) = H0(x). ∂t Next, consider the problem (1) in the model of a random flow with an update: −→ −→ −→ V (t, x) = V k(x), t ∈ [kτ, (k + 1)τ), k = 0, 1, 2, ..., τ > 0 is the update time, V k(x) are independent identically distributed random velocity fields with nonzero means. −→ −→ We denote by H(t, x) = hH (t, x)i the average magnetic field in the distribution of the velocity field. −→ In [1], an equation for H(t, x) was obtained by means of the transition to the Fourier −→ transform H(t, x) with respect to the spatial coordinate. In this paper, we derive the equation for the mean magnetic field directly, using the probabilistic representation of the solution (1) in the form of a conditional mathematical expectation along the trajectories −→ determined by the operator A = νm4−(V, ∇) of the random process. In the asymptotic analysis of the resulting equation, we use the expansions of the coefficients of the equation −→ H(t, x) with respect to the degree of the small update time τ.

Keywords: Random stream with update, a magnetic ﬁeld, average magnetic ﬁeld, conditional mathematical expectation.

2010 Mathematics Subject Classiﬁcation: 60J25, 60J22, 60J55 References: [1] Heydari A., Balakrishnan S.N. Path planning using a novel ﬁnite horizon suboptimal controller, Journal of guidance, control and dynamics, vol. 36, 1210–1214, 2013.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 141

Minisymposium: Actual mathematical problems in control theory

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 142

Projector approach for constructing the zero order asymptotic solution for the singularly perturbed linear-quadratic control problem in a critical case

Galina Kurina1, Nguyen Thi Hoai2 1 Voronezh State University, Russia, Institute of Law and Economics, Russia, and Federal Research Center ”Computer Science and Control” of Russian Academy of Sciences, Russia [email protected] 2 VNU, Hanoi University of Science, Vietnam [email protected]

Abstract: This report is devoted to linear-quadratic optimal control problems with a weak control, fixed left point and the state equation of the form dx (1) ε = A(t, ε)x + εB(t, ε)u + εf(t, ε), t ∈ [0,T ], dt where T > 0 is fixed; ε ≥ 0 is a small parameter; x = x(t, ε) and u = u(t, ε) are m− dimensional and r− dimensional vector functions, respectively; real-valued matrices of suitable size A(t, ε), B(t, ε), and f(t, ε) are sufficiently smooth with respect to both parameters t and ε; moreover, the matrix A(t, 0) is singular. Using the direct scheme method consisting in immediate substituting a postulated asymptotic solution into the problem condition and receiving problems for finding asymptotic terms, the zero order asymptotic solution is constructed under some conditions. In contrast to paper [1], where the problem considered has been studied, the projector approach is applied here. This approach allows us to make the algorithm of constructing the asymptotic solution more clear and to correct some inaccuracies in [1].

Keywords: linear-quadratic optimal control problems, singular perturbations, weak controls, critical case, asymptotic expansions, boundary-layer functions, direct scheme method

2010 Mathematics Subject Classiﬁcation: 49J15, 41A60, 34E15 References: [1] N. T. Hoai, Asymptotic solution of a singularly perturbed linear-quadratic problem in critical case with cheap control, J Optim Theory Appl, vol. 175, no 2, 324–340, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 143

A Solution to Inverse Problems For Dynamic Control Systems 1 Nina Nikolaevna Subbotina1,2, Evgeniy Aleksandrovitch Krupennikov1,2 1 N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia 2 Ural Federal University named after the ﬁrst President of Russia B.N.Yeltsin, Yekaterinburg, Russia [email protected], [email protected]

Abstract: Inverse problems of reconstruction of realized controls for dynamic control systems linear in controls and non-linear in phase coordinates are considered. It is assumed that inaccurate measurements of the realized trajectory are known. Inverse problems have been studied by many authors. The approach suggested by Yu.S. Osipov and A.V. Kryazhimskii [1] is one of the closest to the presented material. It is originated form the differential games theory. Another method for solving dynamic reconstruction problems by known history of inaccurate measurements has been suggested by N.N. Subbotina, E.A. Krupennikov and T.B. Tokmantsev [2, 3]. A modification of this approach is presented. It relies on necessary optimality conditions for an auxiliary variational problem of finding stationary points of a convexo-concave integral discrepancy functional. The functional is a variation of a Tikhonov regularizator. Results of simulation are exposed.

Keywords: Nonlinear control systems, inverse problem, calculus of variations, hamiltonian system, necessary optimality conditions 2010 Mathematics Subject Classiﬁcation: 49M05,34A55 References:

[1] Kryazhimskii A.V., Osipov Yu.S., Modelling of a control in a dynamic system. In: Engrg. Cyber- netics 21 (2), 38ˆa“47(1984). [2] Subbotina, N.N., Tokmantsev, T.B., Krupennikov, E.A., On the Solution of Inverse Problems of Dynamics of Linearly Controlled Systems by the Negative Discrepancy Method // Proceedings of the Steklov Institute of Mathematics, 2015, Vol. 291, Pleiades Publishing, Ltd., 2015, pp. 253ˆa“262. [3] Subbotina, N.N.,Tokmantsev, T.B., Krupennikov, E.A., Dynamic Programming to Reconstruc- tion Problems for a Macroeconomic Model // IFIP Advances in Information and Communication Technology. [S.l.]: Springer, 2017. Vol 494. P. 472-481.

1This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00074) Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 144

A design of an approximate regulator for a weakly nonlinear terminal control problem using the penalty functions method Mikhal Dmitriev1, Dmitry Makarov2 1 Institute for Systems Analysis, FRC CSC RAS, Russia [email protected] 2 Institute for Systems Analysis, FRC CSC RAS, Russia [email protected]

Abstract: This report is devoted to the application of the penalty functions method to the construction of approximated terminal regulator for the next optimal control problem:

t1 1 Z J(u) = xT Q(x, ε)x + uT Ru dt → min, (1) 2 u t0 0 n r x˙ = A(x, ε)x + B(x, ε)u, x(t0) = x , x(t1) = 0, x ∈ X ⊂ R , u ∈ R , where 0 < ε ≤ ε0 1,Q(x, ε) = Q0 + εQ1(x) ≥ 0,A(x, ε) = A0 + εA1(x),B(x, ε) = B0 + εB1(x). Thus, we consider the problem for a weakly nonlinear state-dependent coeﬃcients (SDC) system. By means of introducing the quadratic penalty function

1 T x (t1)F x(t1), 0 < µ 1, F > 0, 2µ problem (1) becomes a SDC optimal control problem with a free right-hand endpoint. It can be investigated on the basis of the approach proposed in [1]. First of all, using ε and µ matching, we study the asymptotics of the auxiliary equation solution. This equation is close to the diﬀerential matrix SDC Riccati equation obtained in [1]. Then, on the basis of asymptotics, an approximate terminal regulator with extrapolation properties is constructed. This work was supported by the Russian Science Foundation (Project No. 17-11- 01220).

Keywords: terminal control, nonlinear systems, small parameter, state-dependent Ric- cati equations, penalty method.

2010 Mathematics Subject Classiﬁcation: 34E10, 34J15, 49M30, 49N35

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 145

Poster Presentations

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 146

A conjugate gradient method for global optimization via a random perturbations

Raouf Ziadi Department of Mathematics University Ferhat Abbas Setif 1,Setif 19000, Algeria [email protected], [email protected]

Abstract: In this paper a method for solving global optimization problems where the objective function is not necessarily convex is presented. This method is a modification of the Polak-Ribièreconjugate gradient method involving the adjunction of a stochastic perturbation. This approach leads to a stochastic descent method where the deterministic sequence generated by Polak-Ribièreconjugate gradient method is replaced by a sequence of random variables. Mathematical results concerning the convergence to a global minimum are established. Numerical experiments are achieved on some typ- ical test problems, particularly the engineering problem of Lennard-Jones clusters. A comparison with well known methods is carried out to show the performance of our algorithm.

Keywords: Global optimization,Random perturbation, Polak-Ribi`ereconjugate gradient method, Lennard-Jones clusters problem. References: [1] M. Gendreau, J. E. Potvin, Handbook of Metaheuristics, International Series in Operations Re- search & Management Science, vol. 146. Springer, New York, 10,978-1, 2011. [2] Brownlee, J. Clever algorithms: nature-inspired programming recipes. Jason Brownlee, 2011. [3] Pogu, M., Souza, J. E., Global optimization by random perturbation of the gradient method with a ﬁxed parameter. J. Glob. Optim. 5(2),159–180, 1994. [4] Pytlak, R., Conjugate gradient algorithms in nonconvex optimization. Springer–Verlag Berlin Hei- delberg, 2009.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 147

Numerical simulation of magnetohydrodynamic process in metallurgical DC furnaces

Oleg Kazak Ural Federal University, Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ekaterinburg, Russia [email protected]

Abstract: In recent years metallurgical industry is search for modern technology based on low energy and resources consumption. One of modern approach for remelting of raw materials is DC furnaces with bottom electrodes. The exploitation of such furnaces has shown problems with increase wear of the bottom electrodes. The reason of this increase wear is vortex flow generated by electromagnetic forcing in liquid metal. In present work the model of preceding process of liquid metal in DC furnace with bottom electrodes, as well software based on the standard commercial software are developed [1]. This model provide possibility to carry out simulations of electric and magnetic parameters, temperature distribution with Joule heat and heat from the arc and hydrodynamic process with taking into account electromagnetic parameters as Lorentz force and heat exchange processes. The work deals not only with development of the approaches for the numerical simulation but also advice methods for the control of liquid metal vortex flow in the DC furnace with the bottom electrodes by changes of bottom electrode parameters and furnace volume construction. It is shown that cooling down the bottom electrode to the melting metal temperature leads to the decrease of shear stress on the fettle area by 15 %, bottom electrode lifting above the surface at the electrode radius â“by 30 %, and the bath form change â“by 50 % [2].

Keywords: Numerical simulation, Electrovortex flow, DC metallurgical furnace References: [1] Kazak, O.; Semko, O.: Modelling of electrovortex and heat flows in dc arc furnaces with bottom electrode. Ironmaking and Steelmaking, 2012, DOI: 10.1179/1743281212Y.0000000081 [2] Kazak, O.: Modelling of vortex flows in DC electric arc furnace with different bottom electrode positions Metallurgical and Materials Transactions B, 2013, DOI: 10.1007/s11663-013-9899-4

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 148

Some approaches for numerical modeling of electrovortex ﬂows in a ﬁnite volume

Oleg Kazak Ural Federal University, Department of Theoretical and Mathematical Physics, Laboratory of Multi-Scale Mathematical Modeling, Ekaterinburg, Russia [email protected]

Abstract: Electro Vortex flow is very important in various applications such as electrical metallurgical furnace, electrical welding, cristal growth, etc. This work presented investigation of electrovortex flows of a liquid metal in a finite volume for various electromagnetic and hydrodynamic regimes. The melt motion was described by magnetic hydrodynamic equations for an incompressible isothermal and non-isothermal fluid. A strategy for solving the problem with the use of standard packages of applied programs has been developed. The laminar flow of the melt at small Reynolds numbers, for which there is a known solution, as well as the turbulent flow, for which there are experimental data, have been calculated. In calculating the turbulent flow, various turbulence models were used. It has been shown that the k-, k-ω and LES models describe satisfactorily the electrovortex flow of the liquid metal and agree well with experimental data. The good agreement between the results of calculations made with the use of various methods and software packages, as well as between theoretical and experimental data, provide the reliability of the developed methods and approaches and the validity of the obtained results.

Keywords: Lorentz force, turbulent electrovortex flow, numerical simulation References: [1] V.V. Boyarevich, Ya.Zh. Freiberg, E.I. Shilova, and E.V. Shcherbinin, Electrovortical Flows [in Russian], Zinatne, Riga 1985 [2] V.G. Zhilin, Yu.P. Ivochkin, A.A. Oksman, I.O. Teplyakov, and S.N. Vavilov, Investigation of thermal and hydrodynamic effects accompanying the electric current spreadings in the bulk of a liquid metal, Heat and Mass Transfer MIF-2008: 6th Minsk Int. Forum, Minsk, 1923 May 2008, ITMO im. A.V. Lykova, Minsk 2008 [3] O.V. Kazak A.N. Semko Numerical modeling of electro-vortical flows in a confined volume Septem- ber 2012Journal of Engineering Physics and Thermophysics 85(5) DOI10.1007/s10891-012-0759-x

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 149

Nonstationary Growth of Spherical Particles in a Supercooled Melt

D.V. Alexandrov, E.V. Makoveeva, A.A. Ivanov, I.O. Starodumov Ural Federal University, Russian Federation [email protected]

Abstract: The processes of industrial production of many materials are frequently accompanied by phase transformations of the liquid-solid type. Such phase transitions often occur in the presence of a supercooled region - a two-phase zone. This zone is filled with crystals of solid phase, evolving in a metastable liquid. Depending on the supercooling, the crystals can be either microscopic aggregates or dendritic-like structures [1]. The evolution of an ensemble of microscopic particles in a supercooled melt is usually described using the kinetic equation for the particle-size distribution function and the balance equation for the temperature or supercooling of the liquid. The mathematical formulation of the problem depends on the law for the rate of growth of individual particles evolving in a supercooled medium. Traditionally, the steady-state approximation is used for this velocity, which follows from the solution of the stationary Stefan thermal problem for a spherical particle [2]. However, the release of heat on the surfaces of growing crystals significantly changes the temperature field around them and leads to the need to consider the non-stationary process of their growth. In this paper, the main nonstationary corrections to the steady-state law for the growth rate of spherical crystals in a supercooled metastable liquid are determined.

Keywords: Moving boundary problem, partial diﬀerential equations, Stefan problem

2010 Mathematics Subject Classiﬁcation: 82C26, 35R37, 80A20 References: [1] D.V. Alexandrov, P.K. Galenko, Dendrite growth under forced convection: analysis methods and experimental tests, Physics-Uspekhi, vol. 57, 771–786, 2014. [2] D.V. Alexandrov, A.P. Malygin, Transient nucleation kinetics of crystal growth at the intermediate stage of bulk phase transitions, J. Phys. A: Math. Theor., vol. 46, 455101, 2013.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 150

On the Theory of Two-Phase Zone Incipience due to the Eﬀect of Constitutional Supercooling

D.V. Alexandrov, I.V. Alexandrova, A.A. Ivanov, O.V. Gusakova, L.V. Toropova, E.A. Titova Ural Federal University, Russian Federation [email protected]

Abstract: Directional crystallization of binary melts is one of the ways of obtaining solid materials, the quality of which is completely determined by the physical and regime parameters of the solidification process. It is well known that, under certain conditions, the solid- liquid interface becomes morphologically unstable [1] and the constitutional supercooling occurs ahead of the planar solidification front [2], which leads to the appearance of a metastable region. In this supercooled region, crystals can grow in the form of dendrites, grow on impurity inclusions, etc. As this zone is transitional between the already formed crystal and the melt, it is called a two-phase zone. In this paper, we present the results of numerical experiments carried out in accordance with the model taking into account the formation of a two-phase zone. In particular, qualitative and quantitative patterns of the time dependence of the formation of a two-phase zone in the Fe-Ni melt are obtained under two different regimes of active and passive cooling. Based on the results of calculations, it can also be concluded that, in both cases, there are such values of the cooling parameters, under which a two- phase zone is formed very close to the right boundary of the sample, i.e. practically the whole sample remains homogeneous. Choosing the appropriate cooling mode and its parameters makes it possible to optimize the metallurgical process both in the quality of the obtained alloys and in the speed of their production.

Keywords: Moving boundary problem, partial diﬀerential equations, Stefan problem

2010 Mathematics Subject Classiﬁcation: 82C26, 35R37, 80A20 References: [1] W.W. Mullins, R.F. Sekerka, Stability of a planar interface during solidiﬁcation of a dilute binary alloy, J. Appl. Phys., vol. 35, 444–451, 1964. [2] D.V. Alexandrov, On the theory of the formation of the two-phase concentration-supercooling region, Dokl. Phys., vol. 48, 481–486, 2003.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 151

Analytical Description of Directional Crystallization with Two-Phase Regions

I.V. Alexandrova, A.A. Ivanov, D.V. Alexandrov Ural Federal University, Russian Federation [email protected]

Abstract: The present paper is devoted to the construction of an analytical solution of a nonlinear nonstationary directional crystallization model with two moving boundaries. The nonstationary process of directional crystallization of three-component melts (solutions) is investigated in the presence of two phase transition zones located between the pure solid and liquid phases. An analytical solution of the nonlinear system of heat and mass transfer equations is found in the presence of two moving interphase boundaries. The temperature distributions, the concentrations of dissolved impurities, and the fractions of the solid and liquid phases in the two-phase zones are determined. The laws of motion of the boundaries between two-phase zones are found too. We consider the heat and mass transfer equations in the moving two-phase zone ∂Tp ∂ ∂Tp ∂ϕA (1) ρc = k(χ) + LV , ∂t ∂z ∂z ∂t ∂B ∂ ∂B ∂ϕA ∂C ∂ ∂C ∂ϕA (2) χ = DB χ + B , χ = DC χ + C , ∂t ∂z ∂z ∂t ∂t ∂z ∂z ∂t where Tp is the temperature, B and C are the concentration of impurity, k = klχ + ks(1 − χ), ρ, c, kl, ks, DB and DC represent the constant coeﬃcients. The solid phase fractions ϕA, ϕB and the liquid phase fraction χ are the time-dependent functions of the spatial coordinate z. This problem has been solved on the basis of techniques developed in our previous works [1,2].

Keywords: Moving boundary problem, partial diﬀerential equations, self-similar variables

2010 Mathematics Subject Classification: 82C26, 35R37, 80A20 References: [1] D.V. Alexandrov, Solidification with a quasiequilibrium mushy region: exact analytical solution of nonlinear model, Journal of Crystal Growth, vol. 222, 816–821, 2001. [2] D.V. Alexandrov, I.G. Nizovtseva, A.P. Malygin, H.-N. Huang, D. Lee, Unidirectional solidification of binary melts from a cooled boundary: analytical solutions of a nonlinear diffusion-limited problem, Journal of Physics: Condensed Matter, vol. 20, 114105, 2008.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 152

Nonlinear Dynamics of Unsteady-State Crystallization of a Binary System

I.V. Alexandrova, D.V. Alexandrov, A.A. Ivanov, A.P. Malygin Ural Federal University, Russian Federation [email protected]

Abstract: Quite often, when the materials solidify between a purely liquid and a purely solid phase, a zone of a two-phase state of matter (liquid and solid) is formed. Its formation is a reaction of the system to the onset of instability of a planar front and the appearance of constitutional supercooling. Equations describing the processes of heat and mass transfer in a two-phase zone are complex nonlinear partial differential equations. The boundary conditions for these equations should be formulated at two moving boundaries, the positions of which are unknown. Thus, the problem of determining the fields of temperature and concentration in a two-phase zone, its dimensions and position, is a complex nonlinear system of thermophysical equations with moving boundaries [1,2]. The present paper is concerned with analytical description of self-similar crystallization regime in the presence of a phase transition layer (two-phase zone) filled with the growing solid phase and liquid. A nonlinear heat and mass transfer model is simplified to describe the case of small variations of the solid phase fraction in the two-phase zone. An exact analytical solution of simplified model is obtained. We show that this solution is in good agreement with the previously known theory. An important point is that the obtained solution has a wider scope of applicability than the asymptotic solution previously found in the vicinity of the bifurcation point.

Keywords: Moving boundary problem, partial diﬀerential equations, self-similar variables, two-phase layer

2010 Mathematics Subject Classification: 82C26, 35R37, 80A20 References: [1] A.C. Fowler, The formation of freckles in binary alloys, IMA Journal of Applied Mathematics, vol. 35, 159–174, 1985. [2] D.V. Alexandrov, D.L. Aseev, Directional solidification with a two-phase zone: thermodiffusion and temperature-dependent diffusivity, Computational Materials Science, vol. 37, 1–6, 2006.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 153

How the convective heat transport at the solid/liquid phase interface inﬂuences the stable mode of dendritic growth

L.V. Toropova1, D.V. Alexandrov1, P.K. Galenko2 1Ural Federal University, Russian Federation [email protected] 2 Friedrich-Schiller-Universit¨atJena,Germany

Abstract: Convection plays an essential role in the growth of dendrites. Convection may influence the transport of heat and substances as well as mechanical deformation of dendritic crystals. In this paper, the effect of convective heat transport that substantially changes the dendrite tip diameter and its tip velocity is demonstrated. In addition, the Gibbs- Thomson condition connecting the phase transition temperature has been taken into account with allowance for the kinetic contribution arising from the effect of attachment kinetics at the phase transition boundary [1]. In this paper the growth of an anisotropic dendrite under conditions of convective heat transfer at the solid-liquid interface was analyzed theoretically. An analytical solution is obtained for the temperature distribution in the liquid phase for the parabolic and paraboloidal forms of the needle-like dendrite. The stability analysis is carried out and the integral of microscopic solvability is defined [2]. The selective ratio for the stable growth rate of the dendritic tip and its diameter is derived.

Keywords: Dendritic growth, Phase transitions, Solvability condition, Convective heat and mass transfer

2010 Mathematics Subject Classiﬁcation: 82C26, 35R37, 80A20 References: [1] D.V. Alexandrov, P.K. Galenko, Dendrite growth under forced convection: analysis methods and experimental tests, Physics-Uspekhi, vol. 57, 771–786, 2014. [2] D.V. Alexandrov, P.K. Galenko, Thermo-solutal and kinetic regimes of an anisotropic dendrite growing under forced convective ﬂow, Phys. Chem. Chem. Phys., vol. 17, 19149-19161, 2015.

Stability anlaysis of conformable fractional order nonlinear systems Ghania Rebiai1,1, Abdellatif Ben Makhlouf 2,2, Mohamed Ali Hammami 2, Assia Guezane-Lakoud3 1 Department of Mathematics,University of Guelma, Guelma, Algeria 2Department of Mathematics, Faculty of Sciences, Sfax University, Tunis 3Department of Mathematics, Faculty of Sciences, Badji Mokhtar University, Annaba, Algeria E-mail: 1rebeai [email protected], [email protected], a [email protected] Abstract: In this paper, we study the uniformly fractional exponential stability for some class of systems like class of perturbed systems and class of nonlinear fractional- order equations with control using the Lyapunov-like function.

Keywords: Conformable fractional derivative, fractional practical exponential stability.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 154

A new suﬃcient conditions on the generalized spectrum method to deal with spectral pollution Ammar Khellaf 1 1University of Guelma, Universit 8 Mai 1945 Guelma, Algeria

E-mail: [email protected]

Abstract: In this work we solve the spectral pollution. We suggest a modern method based on generalized spectral techniques, where we show that the propriety L is hold with norm convergence. We describe the theoretical foundations of the method in details, as well as we illustrate its eﬀectiveness by numerical results. It is well known that the spectral pollution is the weakness of projection methods for an unbounded operator. Thus, our technic is an alternative method based on disrupting an unbounded operator by a bounded one until that the spectral properties transform on controlled case. The generalized spectrum takes its place as the desired case. We will show that every unbounded operator contains a decomposition of two bounded operators which carry all the spectral properties. Through this decomposition and basing upon its numerical approximations the phenomenon of spectral pollution will be resolved. The natural framework of our research is a complex separable Banach space (X, ||.||). Our numerical results are applied on harmonic oscillator operator, which is deﬁned over 2 L (R) by (1) Au = −u00 + x2u.

Generalized spectrum. Let T and S two operators in BL(X), we define the generalized resolvent set by (2) re(T,S) = {λ ∈ C :(T − λS) is bijective}. The generalized spectrum set is Sp(T,S) = C \ re(T,S). For z ∈ re(T,S), we define R(z, T, S) = (T − zS)−1, the generalized resolvent operator. We define λ ∈ C as a generalized eigenvalue when (T − λS) is not injective, then the set E(λ) = Ker(T −λS) is the generalized spectral subspace. We say that λ has an finite algebraic multiplicity if there exist α where dim Ker(T − λS)α < ∞. If the operator S is invertible, we have Sp(T,S) = Sp(S−1T ).

−1 Theorem 8.1. Let λ ∈ re(T,S) and µ ∈ C where |λ − µ| < ||R(λ, T, S)S|| , then µ ∈ re(T,S).

Theorem 8.2. The set Sp(T,S) is closed in C.

Theorem 8.3. The function R(·,T,S): re(T,S) → BL(X) is analytic, and its derivative given by R(·,T,S)SR(·,T,S).

References: [1] E.B. Davies and M. Plum, Spectral Pollution, arXiv:math/0302145v1, (2002) [2] H. Guebbai, Generalized Spectrum Approximation and Numerical Computation of Eigenvalues for Schr¨odinger’sOperators 45,60 (Accepted in Lobachevskii Journal ofMathematics 2013).

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 155

On the Stability of High-Performance Crystal Growth Simulations by the MPFC Method

Ilya Starodumov, Dmitri Alexandrov, Evgeny Pavlyuk Laboratory of Multi-scale Mathematical Modeling, Department of Theoretical and Mathematical Physics, Ural Federal University, 620000 Ekaterinburg, Russia [email protected]

Abstract: Computer modeling of crystal growth processes is often associated with complex numerical calculations. The results of such calculations cannot always be predicted. In some cases, the features of mathematical methods of calculation in conjunction with the assumptions of mathematical models can lead to the appearance of unexpected results. Such results on the one hand will be correct from a mathematical point of view, but on the other hand they will not have physical justification and experimental confir- mation. Obviously, in such cases it is necessary to carefully process the results obtained taking into account the physical meaning and to avoid false conclusions. And it is especially important to take this into account in the case of large-scale high-performance simulations. Earlier, high-perfomance computational algorithm for the modified phase field crystal model (MPFC) of crystal growth were formulated [1] and the results of some three-dimensional simulations are obtained [2]. In current paper we present some unstable results of simulations by the MPFC and discuss such a phenomenon.

Keywords: Mathematical modeling, simulations, crystal growth, phase ﬁeld crystal

2010 Mathematics Subject Classification: 35Q35,35Q86 References: [1] J. Bueno, I. Starodumov, H. Gomez, P. Galenko, D. Alexandrov, Three dimensional structures predicted by the modified phase field crystal equation, Computational Materials Science, 111, 310–312, 2016. [2] I. Starodumov, V. Ankudinov, P. Galenko, Simulation of crystalline pattern formation by the MPFC method, MATEC Web Conf., 129, 02035, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 156

Algorithm of solution of a linear matrix inhomogeneous diﬀerence equation with small step

Zamirbek Imanaliev1, Zhyrgalbubu Barakova2, Beyshembek Ashirbaev3 1 Kyrgyz State Technical University after named I. Razzakov [email protected] 2 Institute of Electronics and Telecommunication at Kyrgyz State Technical University after named I. Razzakov

janna05−[email protected] 3 Kyrgyz State Technical University after named I. Razzakov [email protected]

Abstract: There is an algorithm for solving a linear matrix inhomogeneous diﬀerence equation is proposed in this paper. Also the conjugacy conditions of the homogeneous matrix diﬀerence equation and the ad joint equation to this equation are proved.

Keywords: A singularly perturbed differential equation, a system with a small parameter, a simple structure matrix, multiple small steps. References: [1] Imanaliev Z.K. On a method of optimal control for singularly perturbed systems with minimal energy // Vestnik KNU im. Zh. Balasagyn. 2005. Issue 3, Series 5. C. 25-30 [2] Imanaliyev ZK, Ashirbaev B.Y. Derivation of one of the controllability criteria for singularly perturbed systems of optimal control // Izvestiya KGTU im. I. Razzakov. 2006. â,,–9. C.5-10. [3] Imanaliev Z.K., Ashirbaev B.Y. Algorithm for solving a linear matrix difference equation with a small step // Problems of modern science and education. Ivanovo, the Russian Federation, 2016. â,,–8 (50). C.8-10. [4] Ashyrbaev B.Y. Decomposition and an algorithm for solving optimal control problems with a small step // Izvestiya KGTU them. I.Razzakov. 2016. â,,–3 (39). P.25-31. [5] Vasilyeva AB, Butuzov V.F. Asymptotic expansions of solutions of singularly perturbed equations. M: Science, 1973. - 272 p. [6] Andreev Yu.N. Control of finite-dimensional linear objects. - Moscow: Nauka, 1976.- 424p.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 157

About one inverse problem of time fractional evolution with an involution perturbation

Baurzhan Aibek1, Aizat Aimakhanova1,2, Gani Besbaev1,3, Makhmud Sadybekov1 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, 2 Asfendiyarov Kazakh National Medical University, Kazakhstan 3 Auezov South Kazakhstan State University, Kazakhstan [email protected], [email protected], [email protected], [email protected]

Abstract: We consider an inverse problem for a one-dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. We will consider a process which is so slow that it is described by an evolutionary equation with a fractional time derivative. Thus, this process is described by equation α (1) D∗ Φ(x, t) − Φxx (x, t) + εΦxx (−x, t) = f (x) in the domain Ω = {(x, t): −π < x < π, 0 < t < T }. Here f(x) is the influence of an external source that does not change with time; t = 0 is an initial time point and t = T α is a final one; and the derivative D∗ defined in [1] as α α D∗ u (x, t) = D0t u (x, t) − u (x, 0) − tut(x, 0) , 1 < α < 2 is a Caputo derivative for a regular function built on the Riemann-Liouville derivative α D0t. Such a Caputo derivative allows us to impose initial conditions in a natural way. As the additional information we take values of two initial and one final conditions of the temperature

(2) Φ (x, 0) = φ (x) , Φt (x, 0) = ρ (x) , Φ(x, T ) = ψ(x), x ∈ [−π, π].

Since the wire is closed, it is natural to assume that the temperature at the ends of the wire is the same at all times: (3) Φ (−π, t) = Φ (π, t) , t ∈ [0,T ] .

Consider a process in which the temperature at one end at every time point t is proportional to the (fractional) rate of change speed of the average value of the temperature throughout the wire. Then, Z π α (4) Φ (−π, t) = γD∗ Φ(ξ, t) dξ, t ∈ [0,T ] . −π Here γ is a proportionality coefficient. Thus the investigated process is reduced to the following mathematical inverse problem: Find a right-hand side f(x) of the subdiffusion equation (1), and its solution Φ(x, t) subject to the initial and final conditions (2), the boundary condition (3), and condition (4). This research is financially supported by a grant AP05133271 and by the target program BR05236656 from the Science Committee from the Ministry of Science and Education of the Republic of Kazakhstan.

Keywords: Inverse problem; fractional evolution equation; equation with involution; periodic boundary conditions; method of separation of variables.

2010 Mathematics Subject Classiﬁcation: 35K20, 35L15, 35R11, 35R30 References: [1] B. Ahmad, A. Alsaedi, M. Kirane, R.G. Tapdigoglu; An inverse problem for space and time fractional evolution equations with an involution perturbation, Quaestiones Mathematicae, vol. 40, no. 2, 151-160, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 158

Canonical transformation and feedback linearization of state-dependent coeﬃcient nonlinear systems

Aleksey Kabanov FRC CSC RAS, Russia, and Sevastopol State University, Russia South Kazakhstan State Pedagogical Institute, Kazakhstan [email protected]

Abstract: This report is devoted to a feedback linearisation problem for a nonlinear system with state-dependent coeﬃcients:

(1)x ˙(t) = A(x)x + B(x)u, x(0) = x0, x ∈ Rn, u ∈ R1. Here A(x),B(x) are state-dependent matrices of the dimensions n × n, n × 1 correspond- ingly. The problem is to ﬁnd a non-singular similarity transformation z = T (x)x such that the system (1) can be transformed to a quasi-linear canonical form

(2)z ˙(t) = Ac(z)z + Bc(z)u, z(0) = z0, where Ac(z),Bc(z) is a canonical pair in the form of Brunovskiy. This canonical transformation signiﬁcantly simplify feedback linearization problem for transformed system (2), and knowing the inverse transformation x = T −1(z)z, one can obtain a feedback linearization control for original system (1). There are several techniques for designing such canonical transformation [1,2]. In this paper we introduce another approach based on the concept of state-dependent transfer function and controllability matrix for nonlinear system (1).

Keywords: state-dependent coeﬃcient, nonlinear system, feedback linearisation, canonical transformation

2010 Mathematics Subject Classiﬁcation: 35J05, 35J08, 35J25 References: [1] A.A. Kabanov, Full-state Linearization of Systems via Feedback Using Similarity Transformation, 2016 International Siberian Conference on Control and Communications (SIBCON 2016). Pro- ceedings. National Research University Higher School of Economics. Russia, Moscow, May 12-14, 2016. [2] A.A. Kabanov, Feedback Linearization of Continuous and Discrete Multidimensional Systems, Mekhatronika, Avtomatizatsiya, Upravlenie, 2017, vol. 18, no. 6, pp. 363-370.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 159

Strong approximation by Fourier series and generalized periodic Nikol’skij-Besov-Morrey spaces

Zhuldyz Baituyakova1, Dauren Matin1, Meruert Ilyasova1 1L.N.Gumilyov Eurasian National University, Kazakhstan [email protected], [email protected]

Abstract: In recent years, a lot of attention has been paid to study of Morrey type spaces. Many applications in partial diﬀerential equation of Morrey spaces and Nikol’skij-Besov spaces have been given in work G. Di Fazioand, M. Ragua and the book of T. Mizuhara. The theory of generalized Nikol’skij-Besov-Morrey spaces is developed. Generalized Morrey spaces, with T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function. In the present paper we study Nikol’skij-Besov spaces on scale generalized periodic Morrey space. Finally, we get an interesting characterization s of the space Nϕ,p,q(T) by using diﬀerences of partial sums of the Fourier series and consequence for strong approximation of Fourier series on Morrey space.

2010 Mathematics Subject Classiﬁcation: 35J05, 35J08, 35J25 References: [1] T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Anal., Proc. Conf., Sendai/Jap. 1990, ICM-90 Satell. Conf. Proc., 183-189, 1991. [2] E. Nakai, Hardy “Littlewood maximal operator, singular integral operators, and the Riesz potential on generalized Morrey spaces, Math. Nachr. 166 (1994), 95-103. [3] E. Nakai, A characterization of pointwise multipliers on the Morrey spaces, Sci. Math. 3 (2000), 445-454. [4] S. Nakamura, T. Noi, Y. Sawano, Generalized Morrey spaces and trace operator, Science China Math. DOI 10.1007/s11425-015-5096-z. [5] Baituyakova Zh., Baituyakov Sh, Ilyasova M, On the boundedness of the periodic Hilbert transform on generalized periodic Morrey spaces, AIP Conference Proceeding 1759, 020131(2016); doi: 10.1063/1.4959765 [6] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. PDE 19 (1994), 959-1014. [7] A. Mazzucato, Function space theory and applications to non-linear PDE. Trans. Amer. Math. Soc. 355 (2003), 1297-1369. [8] Zh.Baituyakova, W.Sickel, Strong summability of Fourier series and generalized Morrey spaces, Analysis Math., 43 (3), (2017), 371-414. [9] H.-J. Schmeißer and W. Sickel, On strong summability of multiple Fourier series and smoothness properties of functions, Anal. Math 8 (1982), 57-70. [10] M. Sautbekova and W. Sickel, Strong summability of Fourier series and Morrey spaces, Analysis Math. 40 (2014), 31-62.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 160

Fourier method approach in mixed problems for the heat equation with involution perturbation Sarsenbi Abdisalam 1 1 M. Auezov South Kazakhstan State University, Shymkent, Kazakhstan [email protected]

Abstract: In this paper we consider a mixed problem for the parabolic equation with involution of the form

ut (x, t) = uxx (x, t) − αuxx (−x, t) − q (x) u (x, t) , −1 < x < 1, t > 0; (1) u (0, x) = ϕ (x) , −1 ≤ x ≤ 1; u (−1, t) = 0, u (1, t) = 0, t ≥ 0. (2) In order to solve the problem we apply the separation of variables supposing u(x, t) is a product: u (x, t) = X (x) T (t) . Here the function X (x) gives the solution to the boundary value problem

−X00 (x) + αX00 (−x) + q (x) X (x) = λX (x) , −1 < x < 1, |α| < 1, X (−1) = 0,X (1) = 0. (3) q 1−α Theorem 1. If the number 1+α is not even then the system of eigenfunctions of the spectral problem (3) forms the basis in L2(−1, 1).

Theorem 2. Let the following conditions be satisﬁed: 1) a real continuous function q (x) is non-negative;

q 1−α 2) the number 1+α is not even (−1 < α < 1);

3) the initial condition ϕ (x) of the mixed problem (1), (2) satisﬁes the conditions ϕ (−1right) = ϕ (1 ) = 0 , Then the solution to the mixed problem (1), (2) exists, is unique and can be represented by the series

∞ X −λkt u (x, t) = Ake Xk (x). k=1 This work was supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan, grant AP05131225. References: [1] A.M. Sarsenbi; The theorem on the basis property of eigenfunctions of second order differential operators with involution, AIP Conference Proceedings, vol. 1759, p. 020030, doi: 10.1063/1.4959644, 2016. [2] L.V. Kritskov, A.M. Sarsenbi; Riesz basis property of system of root functions of second-order differential operator with involution, Differential Equations, vol. 53. no. 1. pp. 33-46, 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 161

Modeling of quadratic programming of the mutual inﬂuence of elements of a dynamic series on examples of small agricultural formations

Aiman Iskakova1, Saule Burgumbayva1, Dzhamilya Djumabaeva1, Farabi Zhaksygali1 1 L. N. Gumilyov Eurasian National University, Astana, Kazakhstan

[email protected], [email protected], jamilya−[email protected], [email protected]

Abstract: Proposed model effectively takes into account the mutual influence of the elements of the dynamic series that influence the increase of competitiveness of small agricultural formations, that is, the influence of various economic parameters on each other when they simultaneously manifest themselves. In this case, the forecasting operator is actually trained on the statistical material of the past. From this point of view, the prediction model introduced by us is a neural network.

Keywords: Quadratic Programming, Dynamic Series, Forecasting Operator

2010 Mathematics Subject Classiﬁcation: 90C20, 90C39

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 162

Development of the method and calculation of the heat transfer coeﬃcient of the soil

Rysbaiuly B.1,a, Akishev T.2, Mukhametkaliyeva N.1,b 1 International Information Technology University, Kazakhstan 2 Engineering and Technical University of Ekibastuz, Kazakhstan a [email protected], b [email protected]

Abstract: In this work an approximate method for determining the coefficients of heat transfer of soil is developed. The limitation of approximate value of the heat transfer coefficient and the monotony of the functional are proved. Algorithms for calculating the heat transfer coefficients of soil are developed and software product is designed. Using the measured data the heat transfer coefficient of rock is determined in the different areas of the open mine Ekibastuzsky. Nowadays one of the most relevant problems is to search and elaborate energy saving actions on creation of heat and technological processes with minimum thermal losses. Prior knowledge of thermo physical characteristics of the used and developed structural, heat insulating and lining materials has a huge impact. Thermal characteristics of enclosing constructions significantly influence the thermal and air conditions of the building and the work of heating system. Problems of energy saving and decrease in heat losses significantly influences an ecological situation, technical and economic indicators and capital expenditures of structural objects. In order to solve these problems it is necessary to know thermal capacity, heat transfer, heat conductivity and heat diffusivity of materials. Some materials have passport data, but others do not. Besides actual characteristics of structural materials can change during the operation and do not to correspond to their certificate or the passport. Therefore during the construction of objects of different function it is necessary to be able to define thermal characteristics of materials. A very effective characteristic of the heat exchange process is the heat transfer coefficient h, between the air flow and the surface of soil. A lot of researches reveals the effect on the given dependence of the porosity of the material, the size of a form of the grains, and the effect of all possible physical characteristics of the material (for example, heat capacity, thermal conductivity). There is a huge amount of empirical formulas. But both the data and the formulas that form the basis for them differ sharply at various authors. There are a lot of reasons of a divergence: the influence of the internal resistance of the pieces; Complex geometric structure of the layer, which changes from various factors; the porosity of the layer, and the roughness of the grains. The value of the heat transfer coefficient is strongly dependent on the amount, thickness and thermophysical characteristics of soil layers. In the depth of the cut, values of the heat transfer coefficient decreases. The coefficient has a significant difference at wind speed. References: [1] Chudnovsky A.F. Thermophysics of Soil.1976. M. Nauka p.352. [2] John H.Lienhard, Heat Transfer Textbook, Third Edition, Cambridge, 2008, p.672. [3] Rysbai I B,. Akishev TB Convergence of the iterative method for determining the heat transfer coefficient. Proceedings of the sixth meeting of the Russian-Kazakhstan Working Group on Computing and Information Technologies. KazNU (16-19 March 2009) p. 273-276

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 163

On the Well-Posedness of the Nonlocal Boundary Value Problem for the Diﬀerential Equation of Elliptic Type

Allaberen Ashyralyev1, Ayman Hamad2 1 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Peoples’ Friendship University of Russia (RUDN University), Moscow , Russia. Institute of Mathematics and Mathematical Modeling 050010, Almaty, Kazakhstan. [email protected] 2 Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey. Department of Mathematics, Omar Al-Mukhtar University, El-Beida, Libya. [email protected]

Abstract: In the present paper the abstract nonlocal boundary value problem for the second order diﬀerential equation

T Z −v00(t) + Av(t) = f(t) (0 ≤ t ≤ T ), v(0) = v(T ) + ϕ, v(s)ds = ψ 0 in an arbitrary Banach space E with the positive operator A is considered. The well- posedness of this problem in various Banach spaces is established. In applications, the coercive stability estimates in H¨oldernorms for the solutions of the mixed type nonlocal boundary value problems for elliptic equations are obtained.

Keywords: elliptic equation; positive operators; Well-posedness; coercive stability.

2010 Mathematics Subject Classification: 35J25, 47E05, 34B27 References: [1] A. Ashyralyev, P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Op- erator Theory: Advances and Applications, vol.148, Birkhauser, Verlag, Basel, Boston, Berlin 2004. [2] A. L. Skubachevskii, Elliptic functional differential equations and applications, Birkhauser Verlag, Basel, Boston, Berlin, 1997. [3] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhauser Verlag, Basel, Boston, Berlin,1995.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 164

Error a posteriori estimation for a problem of optimal control of the obstacle

Mohamed Gouasmia1, Radouen Ghanem2 1 University of Applied Mathematics, Annaba, Algeria [email protected] 2 University of Applied Mathematics, Annaba, Algeria radouen [email protected]

Abstract: This poster is devoted to a problem of the optimal control of the obstacle represented as follows

Z Z 1 2 2 (1) min (Tg(ψ) − z) dx + (∇ψ) dx ψ∈Uad 2 Ω Ω subject to: (2) ∀v ∈ K(ψ): a(y, v − y) + hg(y) − f, v − yi ≥ 0 ; y ∈ K(ψ) such that 1 (3) K(ψ) = y ∈ H0 (Ω); y ≥ ψ, pp dans Ω

n  n  X X Z ∂u ∂v Z ∂u Z (4) a(u, v) =  aij dx + bi v dx + cuvdx ∂x ∂x ∂x i=1 j=1 Ω i j Ω i Ω ∞ 0,1 where ai,j , bi and c in L (Ω). Moreover, on suppose ai,j ∈ C (Ω) ( space of Lipschitz functions continues on Ω ) and c non negative, where the form bilinear a(., .) is continuous and coercive with M et m are respective

Despite the non-diﬀerentiability of the y = Tg(ϕ) application, this type of problem has been widely studied by several authors on a mathematical level (see for example [1,2] but little has been done digitally. After ﬁnite element discretization, and in spite of the non-convex character of this kind of problem, our main goal is to give estimates of the a posterior errors on the state and the control (see for example [2]).

Keywords: Probleme of the obstacle, optimal Control , Penalization References: [1] Barbu V., Analysis and Control of Non Linear Inﬁnite Dimensional Systems, Mathematics in Science and Engineering, 190, Academic Press, New York 1993. [2] M. Bergounioux & S. Lenhart, Optimal control of the obstacle in semilinear variationl inqualities, 8: 229-242, 2004.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 165

Time asymptotic behavior of exponential function of Sturm-Liouville operator on the line Okba Gouasmia 1, Akila Diaba 1, Fatma Diaba 1 1 Applied Mathematics Laboratory,Badji Mokhtar University,BP 12, 23000, Annaba, Algeria [email protected]

Abstract: Spectral singularities of Sturm-Liouville operator on the line generate some increasing composants in time asymptotic of the solution of corresponding evolution equation. The calculus of these composants is given using Friedrichs’ model of Sturm- Liouville operator and some scalar function which characterize the point of discontinuity of Fourier transformation of Friedrichs’ model. An example is given.[1]

Keywords: Spectral singularities, Sturm-Liouville operator, Weyl function, Friedrichs’ model, point spectrum, asymptotic behavior.

AMS subject classiﬁcation:47A20, 47A30, 47H12, 46B34 References: [1] E. V. Cheremnikh,Asymptotic behaviour of the solution of some evolution equations, Math. meth. and phys.-mech. ﬁelds, 40(1997), 75–85.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 166

On one class of problems for restoring the density of sources of the process of fractional heat conductivity with respect to initial and ﬁnal temperature

Dina Kusmangazinova1,2, Issabek Orazov1,3, Makhmud Sadybekov1 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, 2 Al-Farabi Kazakh National University, Kazakhstan 3 Auezov South Kazakhstan State University, Kazakhstan [email protected], i [email protected], [email protected]

Abstract: We consider a class of problems modeling the process of determining the temperature and density of sources in the process of fractional heat conductivity with respect to given initial and finite temperatures. Let Ω = {(x, t) , 0 < x < 1, 0 < t < T }. In Ω consider a problem of finding the right-hand side f(x) of the fractional heat equation α (1) D0+ (u (x, t) − u (x, 0)) − uxx (x, t) = f (x) , (x, t) ∈ Ω and its solutions u (x, t) satisfying the initial and final conditions (2) u (x, 0) = ϕ (x) , u (x, T ) = ψ (x) , 0 ≤ x ≤ 1, and the boundary conditions a u (0, t) + b u (1, t) + a u (0, t) + b u (1, t) = 0, (3) 1 x 1 x 0 0 c1ux (0, t) + d1ux (1, t) + c0u (0, t) + d0u (1, t) = 0. α The coefficients ak, bk, ck, dk with k = 0, 1 in (3) are real numbers, D0+ stands for the Riemann-Liouville fractional derivative of order 0 < α < 1: Z t α 1 d y(s)ds D0+y (t) = α , Γ(1 − α) dt 0 (t − s) while ϕ (x) and ψ (x) are given functions. Problems with general boundary conditions with respect to a spatial variable that are not strongly regular are investigated. The problem is considered independently of whether a corresponding spectral problem for an operator of multiple differentiation with not strongly regular boundary conditions has the basis property of root functions. The existence and uniqueness of classical solutions to the problem are proved. The closest to the subject of this paper is [1], in which one case of regular but not strongly regular boundary conditions was considered. This research is financially supported by a grant AP05133271 and by the target program BR05236656 from the Science Committee from the Ministry of Science and Education of the Republic of Kazakhstan.

Keywords: Inverse problem; heat equation; fractional heat conductivity; not strongly regular boundary conditions; method of separation of variables

2010 Mathematics Subject Classiﬁcation: 35K20, 35R30, 35R10 References: [1] Mokhtar Kirane, Salman Amin Malik; Determination of an unknown source term and the temperature distribution for the linear heat equation involving frac-tional derivative in time. (2010). , https://hal.archives-ouvertes.fr/hal-00549130.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 167

Improving Performance of Indoor Visible Light Communication Systems with Hexagonal Designed LED Transmitters Mahmoud Khaled Aburub1, Ahmetjan Bekenov2 1 Cyprus International University, Nicosia, Mersin 10, Turkey

E-mail: [email protected] 2Cyprus International University, Nicosia, Mersin 10, Turkey

E-mail: [email protected]

Abstract: We report a simulation program for indoor visible light communication with hexagonal LED array shape that will allow us to focus the luminance of optical light for the room center with neglecting of the boundary since they will be occupied by furniture. Users will not be in those areas. We are going to model OOK, PPM and QAM modulation techniques with varied semi-angle degrees, as a result we want to prove with hexagonal shape we could make and uniform luminance in the room. We are going to use MATLAB and Simulink for simulation. In this indoor VLC system both LOS and diffuse configurations will be taken into consideration. So, when received optical power at a point calculated, direct path and reflections from the wall should be taken into account as below [1]

PNLEDs R Pr = PtHd(0) + reflections PtdHref (0)

The horizontal intensity at a point (x,y,z) is given as I(0).cos(φ)m Ihor = 2 Dd.cosψ

Keywords: Visible light communications (VLC), LED, illumination, modulation.

References: [1] H. Q. Nguyen et al., ” A MATLAB-Based simulation program for indoor visible light communication system,” in Proc. CSNDSP, Jul. 2010,pp. 537-540.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 168

Approximation of Maxwell’s microscopic boundary conditions for onedimensional nonstationary nonlinear Boltzmann’s twelve moment equations

Auzhan Sakabekov1, Yerkanat Auzhani2, Shinar Akimzhanova3 Satbayev University, Kazakhstan 1 [email protected] 2 [email protected] 3 [email protected] An important problem in aerospace engineering is the prediction of aerodynamic characteristics of aircraft at very high speeds and at high altitudes. In the case of a gas flowing near a solid or inside a region bounded by a solid surface, the boundary conditions describe the interaction of gas molecules with solid walls. The interaction of gas with a solid surface is the source of the drag and the lifting force of the body in the gas flow, as well as the heat transfer between the gas and the solid boundary. The boundary conditions that the particle distribution function must satisfy at the boundary of the region where the particles under study are moving depend on the state of the boundary surface, its temperature, and on the degree of its roughness and purity. Unfortunately, theoretical and experimental studies on the interactions of gas with the surface have been little carried out. The mirror-diffusive Maxwell model [1] is often used as the boundary condition imposed on the distribution function of the gas molecules reflected from the surface of the molecules. To calculate the aerodynamic characteristics of aircraft, the Boltzmann’s moment equations are used. Finite system of moment equations for a specific task with a certain degree of accuracy replaces the Boltzmann equation. It’s necessary, also roughly, to replace the boundary conditions for the particle distribution function by a number of macroscopic conditions for the moments, i.e. there arises the problem of boundary conditions for a finite system of equations that approximate the microscopic boundary conditions for the Boltzmann equation. The question of boundary conditions for a finite system of moment equations can be divided into two parts: how many conditions must be imposed and how they should be prepared. From microscopic boundary conditions for the Boltzmann equation there can be obtained an infinite set of boundary conditions for each type of decomposition. However, the number of boundary conditions is determined not by the number of moment equations, i.e. it is impossible, for example, take as much boundary conditions as equations, although the number of moment equations affects the number of boundary conditions. In addition, the boundary conditions must be consistent with the moment equations and the resulting problem must be correct. The initial and boundary value problem for one-dimensional non-stationary Boltzmann’s equation with boundary conditions of Maxwell was approximated by a corresponding problem for the Boltzmann’s moment system of equations [2]. In this work we approximate the microscopic Maxwell boundary condition satisfied by the particle distribution function for one dimensional Boltzmann’s twelve moment equations and show correctness of the initial and boundary value problem for Boltzmann’s twelve moment system equations.

Keywords: Boltzmann equation, boundary conditions, nonlinear, Boltzmann’s moment system equations, particle distribution functions.

2010 Mathematics Subject Classiﬁcation: 35Q20, 35F60 References: [1] Cercignani C., Theory and application of Boltzmann’s equation. Milano, Italy, 1975. [2] Sakabekov A., Auzhani Y. Boundary conditions for the one dimensional nonlinear nonstationary Boltzmann’s moment system equations - JOURNAL OF MATHEMATICAL PHYSICS 55, 123507 (2014), AIP, USA.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 169

The modeling of heat and mass transfer processes in colloidal suspension

PolinaVinogradova1, Ilya Manzhula2 1 Far East Transport State University,Russia [email protected] 2 Far East Transport State University and Computer Center Far East Branch Russian Academy of Science, Russia [email protected] Abstract: In the current paper, we consider a liquid-phase medium with nanoparticles irradiated by a light beam with a Gaussian intensity profile. As a result of the action of the light field in the medium, gradients of temperature and concentration arise, which cause heat and mass transfer. These phenomena are described by a system of balance equations for temperature and particles both with concentration convection. Let us take into account the fact that the temperature establishment processes go faster than the diffusion processes. Therefore, we study the diffusion processes at a stationary temperature. As a result of introducing dimensionless variables and parametrization, we obtained an initial boundary value problem for a one-dimensional nonlinear parabolic equation. In this case, the intensity with a Gaussian profile was considered as the initial condition. The boundary conditions are obtained by equating to zero the sum of the thermodiffusion and electrostriction flows of nanoparticles. In deriving the parabolic equation, we used a linear dependence of the thermal conductivity of the medium on temperature. Such dependence is confirmed by a number of experiments. We use the predictor-corrector method for the numerical solving of the nonlinear initial boundary value problem. The constructed difference scheme is unconditionally stable and has second-order accuracy in time and space variables. In this paper, we presented the conclusions about the dynamics of concentration using the numerical experiment results. Throughout this note we mainly use techniques from the paper [1].

Keywords: heat and mass transfer, dynamics of nanoparticles, Sore eﬀect, non-linear equation, predictor-corrector method

2010 Mathematics Subject Classification: 35K05, 35K55, 35K57 References: [1] A.I. Livashvili, V.V. Krishtop, Y.M. Karpets, N.M. Kireeva, Formation of bistable medium state in nanofluids under the influence of laser radiation, PROCEEDINGS OF SPIE - THE INTERNA- TIONAL SOCIETY FOR OPTICAL ENGINEERING, ”Asia-Pacific Conference on Fundamental Problems of Opto- and Microelectronics” 2017.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 170

On a parabolic equation with time delay Batuhan Ak¸cın1, Deniz Agirseven1 1 Department of Mathematics, Trakya University, Edirne, Turkey [email protected] [email protected]

Abstract: In this study, exact solution of a delay parabolic equation is found by using separation of variables and Fourier series method [3].We find the numerical solution of initial-boundary value problem for delay parabolic differential equations with Neumann condition by using difference schemes [1-2]. Numerical solutions found by first and second order of accuracy difference schemes are given with tables.

Keywords: Delay parabolic equation, diﬀerence schemes

2010 Mathematics Subject Classification: 65M06, 35K10 References: [1] A. Ashyralyev, D. Agirseven, On convergence of difference schemes for delay parabolic equations, Computers and Mathematics with Applications 66 (2013) 1232-1244. [2] A. Ashyralyev, P. E. Sobolevskii, New Difference Schemes for Partial Differantial Equations, Op- erator Theory Advances and Applications, Birkhauser Verlag, Basel, Boston, Berlin, 2004. [3] I.˙ H. Gürbey, Gecikmeli parobolik bir diferansiyel denklemin ¸cözümleri,Trakya University, Turkey, 2013.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 171

HAM Solution for a delay parabolic equation Barı¸sErk¨ose1, Deniz A˘gırseven1 1 Department of Mathematics, Trakya University, Edirne, Turkey [email protected] [email protected]

Abstract: In this study, exact solution of initial-boundary value problem for a delay parabolic differential equation with Neumann condition is found by using Laplace transform method. By using Homotopy Analysis Method (HAM), we find the numerical solution for this initial-boundary value problem [1]. Comparison of numerical solutions obtained by using HAM and difference schemes is presented in a table [2].

Keywords: Parabolic diﬀerential equation, time delay, homotopy analysis method, Laplace transform

2010 Mathematics Subject Classification: 35K10, 65M06, 65M99 References: [1] I. H. Gürbey, Gecikmeli parobolik bir diferansiyel denklemin ¸cözümleri,Trakya University, Turkey, 2013. [2] A. Ashyralyev, D. Agirseven, On convergence of dierence schemes for delay parabolic equations, Computers and Mathematics with Applications 66 (2013) 1232-1244.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 172

The construction of a solution of a related system of the Laguerre type Zh.N. Tasmambetov1, R.U. Zhakhina2 1 Aktobe Regional State University named after K. Zhubanov, Kazakhstan [email protected] 2 Aktobe Regional State University named after K. Zhubanov, Kazakhstan riscul [email protected]

Abstract: The systems of the Laguerre type, which were obtained from the system of Horn by means of exponential transformation, are considered. The Frobenius-Latysheva method is applied for the construction of the solution. The main theorem on the existence of four linearly independent particular solutions, which are expressed in terms of the 0 degenerate hypergeometric function of M.R. Humbert Ψ2(α, γ, γ ; x, y) in the form of normally-regular series dependent on Laguerre polynomials of two variables, is proved. Formulation of the problem From the system of Horn ) x · Zxx + (γ − x) · Zx − y · Zy − n · Z = 0, (1) 0 y · Zyy + (γ − y) · Zy − x · Zx − n · Z = 0, by means of converting a system of Laguerre type is installed 2  2 x xy 1 2 x · Uxx − xy · Uy + − − + kx + − α · U = 0,  4 2 4  2 . (2) 2 y xy 1 2 y · Uyy − xy · Ux + − − + kx + − β · U = 0,  4 2 4  where k = (α + β + 2 − 2λ)/2 is related with the basic Laguerre system [1]. Theorem is proved. Theorem. The system (2) has four linearly independent partial solutions, which are 0 expressed through the degenerate hypergeometric function of M.R. Humbert Ψ2(α, γ, γ ; x, y) in the form of normal-regular series α + 1 β + 1 x y U(x, y) = exp − − · x 2 · y 2 · Ψ2(−n, α + 1, β + 1; x, y) = 2 2 (3) α + 1 β + 1 x y (α,β) = exp − − · x 2 · y 2 · Ln,n (x, y) 2 2 dependent on the Laguerre’s polynomial of two variables (α,β) Ln,n (x, y) = Ψ2(−n, α + 1, β + 1; x, y).

Frobenius-Latysheva method is used to proof of the theorem . Series of theorems on the necessary condition for the existence of a solution of the form are also proved (3).

Keywords: Related system, the system of the Laguerre type, the system of Horn, normal-regular solution, special curves, rank, anti rank.

2010 Mathematics Subject Classiﬁcation: 34K29, 60H10 References: [1] Zh.N.Tasmambetov, Conﬂuent hypergeometric functions and two variables Laguerre polynomials as solutions of Wilczynski type system, AIP Conference Proceedings 1759, 020137 (2016); cloi: 10.1063/1, 4959751.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 173

Construction of solutions of inhomogeneous systems of Jacobi type Zh.N. Tasmambetov 1, M.Zh. Talipova2 1 Aktobe Regional State University named after K. Zhubanov, Kazakhstan [email protected] 2 Aktobe Regional State University named after K. Zhubanov, Kazakhstan mira [email protected]

Abstract: The possibilities of constructing a general solution of an inhomogeneous system of second-order partial differential equations of Jacobi type, closest to ordinary differential equations of the second order are studied. The method of undetermined coefficients is used for the construction of a particular solution of the system. Specific examples established a connection between systems of Jacobi type with one partial differential equation of the second order obtained by adding two equations of the original system. Formulation of the problem The method of undetermined coefficients is spread for the construction of a general solution of the inhomogeneous system of Jacobi type

x · (1 − x) · Zxx + [γ − (α + 1) x] · Zx + n(α + n) · Z = f1(x, y), (1) 0 y · (1 − y) · Zyy + [γ − (β + 1) y] · Zy + m(α + m) · Z = f2(x, y), where Z = Z(x, y) – the general unknown, fl(x, y)(l = 1, 2) – polynomials. Work on the distribution of this method to inhomogeneous systems of the form (1) was done M.Zh. Talipova [1]. Theorem 1. The general solution of the system (1) is represented as the sum of the general solution Z(x, y) of the corresponding homogeneous system and the particular solution Z0(x, y) of the system (1):

Z(x, y) = Z(x, y) + Z0(x, y). (2)

Theorem 2. Let there be given an inhomogeneous system of Jacobi’s type

x · (1 − x) · Zxx + [γ − (α + 1) x] · Zx + n(α + n) · Z = 21n(n + α) = 7 + 21x + · y, (n − 1)(n + α + 1) 0 (3) y · (1 − y) · Zyy + [γ − (β + 1)y] · Zy + m(α + m) · Z = 21m(m + α) = 7 + 21y + · x. (m − 1)(m + α + 1) Then, the general solution of system (3) is represented as the sum of the general solution Z(x, y) of the of the corresponding homogeneous system and the particular solution Z0(x, y) of the inhomogeneous system (3), that is

Z(x, y) = Z(x, y) + Z0(x, y) = C1Z10(x)Z01(y) + C2Z10(x)Z02(y)+ 7 +C3Z20(x)Z01(y) + C4Z20(x)Z02(y) + − n(n + α) 21 21x 21y − + + . (n − 1)(n + α + 1) (n − 1)(n + α + 1) (n − 1)(n + α + 1)

Keywords: system, particular solution, homogeneous, inhomogeneous, general solution, polynomial, orthogonal.

2010 Mathematics Subject Classiﬁcation: 34K29, 60H10 References: [1] Talipova M.Zh. Construction of normal solutions of inhomogeneous systems of partial diﬀerential equations of the second order, Author’s abstract. ... dis. cand. Almaty, 2007, 12 pp.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 174

Numerical Modeling of a Liquid Droplet Surface Advection: Some Tests of the Modiﬁed VOF Algorithm Nurbulat Jaichibekov1, Bakyt Shalabayeva1, Victor Kireev2 1 L.N. Gumilyov Eurasian National University, Kazakhstan [email protected] 2Bashkir State University, Russia [email protected]

Abstract: The goals of this paper are development and validating of the original computer code based on the algorithm of the Volume-Of-Fluid method [1], which will be the part of the software project for numerical simulation of droplet surface dynamics under the influence of external physical fields (electromagnetic field, in particular). The developed version of the computer code is devoted for tracking of liquid droplet surface in the given velocity field. To verify the efficiency of the code developed and to estimate the accuracy of the droplet surface representation a series of tests was performed. In the first test series, initially the circular and square liquid regions moved at a constant speed along one of the coordinate axes and along the diagonal. In the second test series, the circular and square liquid regions were placed in a circular velocity field and rotated about 360 degrees around the center of rotation as a solid body. In the third test series a solenoidal velocity field was used in which the initially circular shape of the droplet is stretched and twisted into a spiral around the center of rotation, that is the topology of the liquid droplet has been changed. After a while after the calculation begins, the direction of rotation is reversed and the deformed droplet returns to its original position. The numerical simulation results shows that in all the tests the original droplet form is preserved with a sufficient degree of accuracy. In the first and second series of experiments the deviation is of the order of 1%, and in the third series the deviation does not exceed 5%. Thus, the computer code developed adequately simulates the position of the droplet surface. The work is supported by the Ministry of Education and Science of the Republic of Kazakhstan (project AP05134098).

Keywords: Droplet dynamics, Volume-of-Fluid method, numerical modeling

2010 Mathematics Subject Classiﬁcation: 76T99, 65Z05 References: [1] C.W. Hirt, B.D. Nichols, Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, Journal of Computational Physics, vol. 39, no 1, 201–225, 1981.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 175

Two-dimensional calculations of stratiﬁed turbulent ﬂow in a pipe S.K. Matveev1, N.Zh. Jaychibekov2, B.S. Shalabayeva2 1SPbGU, St. Petersburg, Russia [email protected] 2 L.N. Gumilev ENU, Kazakhstan, Astana, Kazakhstan [email protected]

Abstract: In this paper, we consider the stratified turbulent flow of a two-phase medium in inclined pipes. Based on the new turbulence model [1], a program code for calculating two-dimensional flows for the study of two-phase stratified flows in pipes was developed, including taking into account the rough of the pipeline wall. The technique for calculating two-phase flows in extended pipelines is described. The problem of stationary stratified two-phase flow in a pipe of constant cross section in the case of turbulent regime is numerically solved. Calculations of the resistance of a rough pipe are carried out and the results on the influence of roughness on pipe resistance and velocity distribution are presented.

Keywords: stratiﬁed turbulent ﬂow, resistance, two-dimensional calculations, rough surface

47.55.Ca, 2010 Mathematics Subject Classification: 76T10 References: [1] Matveev S.K., Jaychibekov N.Zh., Shalabaeva B.S. Modification of the turbulence model for the calculation of two-phase flow in a pipe. // Scientific journal Herald. Astana: ENU them. L.N. Gumilev, 2017, - â,,–6(121) P.157-161.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 176

Solution of a linear problem of Florin type with the small parameters for the system of the parabolic equations Zhanat K. Dzhobulaeva Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, Al-Farabi Kazakh National University, Almaty, Kazakhstan [email protected] There is studied a linearized two phase free boundary problem of Florin type with the small parameters for the system of the parabolic equations in the Hölderspaces. The original nonlinear free boundary problem of Florin type describes the process of filtration of liquids and gases in the porous medium. There are proved the existence, uniqueness of the solution and uniform coercive estimates of the solution with respect to the small parameters in the Hölderspaces. This research is financially supported by a grant AP05133898 from the Science Com- mittee of the Ministry of Education and Science of the Republic of Kazakhstan.

Keywords: Parabolic equations, small parameters, coercitive estimates, H¨olderspace.

2010 Mathematics Subject Classiﬁcation: 35K20, 35B45, 35B30, 35C15, 35R35

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 177

On construction of the set of comparison functions of the program motion in the probable statement Gulmira Vassilina1, Marat Tleubergenov2 1 Institute of Mathematics and Mathematical Modeling, Kazakhstan, and Almaty University of Energy and Communications, Kazakhstan v [email protected] 2 Institute of Mathematics and Mathematical Modeling, Kazakhstan [email protected]

Abstract: The set of first-order Ito stochastic differential equations and set of comparison functions are constructed. There is stability in probability of the given program motion with respect to comparison functions. Assume that the set (1) Λ : λ ≡ y − ϕ(t) = 0, y ∈ Rn, ϕ ∈ C1, kϕk ≤ l, is given. It is necessary to construct the corresponding set of Ito stochastic differential equations (2)y ˙ = Y (y, t) + σ(y, t)ξ,˙ ξ ∈ Rk, in the class of equations admitting the existence and uniqueness of solution of the equation (2) for the initial conditions y|t=t0 = ϕ(t0). And also we need to construct the set of s-dimensional vector-functions Q(y). With respect to the components of Q(y) there is stability in the probability of the set (1). This stochastic problem is solved by both the Lyapunov functions method [1] and the quasi-inversion method [2]. The given statement of the problem for σ ≡ 0 is considered in [3, 4].

Keywords: stochastic diﬀerential equation, stability in probability, comparison function 2010 Mathematics Subject Classiﬁcation: 34K29, 60H10 References:

[1] G.K. Vasilina, M.I. Tleubergenov, Solution of the problem of stochastic stability of an integral manifold by the second Lyapunov method, Ukrainian Mathematical Journal, 2016, Vol. 68, 1, 14-27, http://dx.DOI 10.1007/s11253-016-1205-6. [2] I.A. Mukhametzyanov, R.G. Mukharlyamov, Equations of program motions, oscow, 1986, 88 p. (in Russian) [3] A.S. Galiullin, Inverse problems of dynamics, Moscow, 1981. 144 p. (in Russian) [4] A.S. Galiullin, Selected proceeding in two volumes, Vol.I, II, Moscow, 2009, 462 p. (in Russian)

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey Fourth International Conference on Analysis and Applied Mathematics 178

Autocorrelation and intercorrelation spectrum to predict of atrial fibrillation recurrence Raquel Cervigón1,Javier Moreno2, Francisco Castells3 1 Escuela Politécnica. Universidad de Castilla-La Mancha (UCLM). Camino del Pozuelo sn. 16071. Cuenca. Spain [email protected] 3 ITACA Institute, Universitat Politécnica de Valencia, Valencia, Spain 2 Javier Moreno Unidad de Arritmias, Hospital Ramóny Cajal, Ctra. de Colmenar Viejo km. 9,100, 28034 Madrid, Spain

Abstract: Atrial fibrillation (AF) is the most common heart rhythm disorder in the world [1] and it is characterized by an uncoordinated atrial electrical activation, whose foci in most cases are found in the pulmonary veins [2]. Frequency domain analysis has been applied to atrial electrograms to quantify the temporal and spatial organization during AF in order to predict the outcome of the three months after a successful ablation intervention. Intra-atrial recordings and pulmonary vein electrogram recordings were obtained before AF ablation procedure were analyzed by spectrum autocorrelation and intercorrelation of atrial activations from dipoles located along different atrial areas to to quantify temporal and spatial organization during AF. Results showed changes in spectral characteristics with higher dispersion in spatial organization along the pulmonary veins in patients that had recurrence in the arrhyth- mia 0.14±0.01 vs. 0.07±0.02 compared with patients that maintained sinus rhythm (p<0.001). The proposed analysis could be useful for a better understanding of electrophysio- logical mechanisms during AF in both groups of patients, where the electrical activity is collected from electrodes located in the pulmonary veins area before the isolation procedure. Keywords: spectrum; correlation; atrial fibrillation; ablation References:

[1] Rahman, Faisal and Kwan, Gene F and Benjamin, Emelia J, Global epidemiology of atrial fibrillation, Nature reviews. Cardiology, vol. 11, no. 11, 639-54, 2014. [2] Skowerski, M and Wozniak-Skowerska, I and Hoffmann, A and Nowak, S and Skowerski, T and Sosnowski, M and Wnuk-Wojnar, A M and Mizia-Stec, K,Pulmonary vein anatomy variants as a biomarker of atrial fibrillation - CT angiography evaluation, BMC cardiovascular disorders, vol. 18, no1, 146, 2018.

Near East University, Lefkosa (Nicosia), Mersin 10, Turkey